Discontinuous Galerkin Timestepping for Nonlinear Parabolic Problems

Discontinuous Galerkin timestepping
for nonlinear parabolic problems
Thesis submitted for the degree of
Doctor of Philosophy
at the
University of Leicester
by
Mohammad Sabawi
Department of Mathematics
University of Leicester
February 2018

“Imagination is more important than knowledge. For knowledge is limited to all
we now know and understand, while imagination embraces the entire world, and
all there ever will be to know and understand.”
Albert Einstein

ii
Discontinuous Galerkin timestepping
for nonlinear parabolic problems
by
Mohammad Sabawi
Abstract
We study space–time finite element methods for semilinear parabolic problems
in (1 + d)–dimensions for d = 2, 3. The discretisation in time is based on the
discontinuous Galerkin timestepping method with implicit treatment of the linear
terms and either implicit or explicit multistep discretisation of the zeroth order
nonlinear reaction terms. Conforming finite element methods are used for the
space discretisation. For this implicit-explicit IMEX–dG family of methods, we
derive a posteriori and a priori energy-type error bounds and we perform extended
numerical experiments. We derive a novel hp–version a posteriori error bounds in
the L∞(L2) and L2(H1
) norms assuming an only locally Lipschitz growth condition
for the nonlinear reactions and no monotonicity of the nonlinear terms. The
analysis builds upon the recent work in [60], for the respective linear problem,
which is in turn based on combining the elliptic and dG reconstructions in [83, 84]
and continuation argument. The a posteriori error bounds appear to be of optimal
order and efficient in a series of numerical experiments.
Secondly, we prove a novel hp–version a priori error bounds for the fully–discrete
IMEX–dG timestepping schemes in the same setting in L∞(L2) and L2(H1
) norms.
These error bounds are explicit with respect to both the temporal and spatial
meshsizes kn and h, respectively, and, where possible, with respect to the possibly
varying temporal polynomial degree r. The a priori error estimates are derived
using the elliptic projection technique with an inf-sup argument in time. Standard
tools such as Grönwall inequality and discrete stability estimates for fully discrete
semilinear parabolic problems with merely locally-Lipschitz continuous nonlinear
reaction terms are used. The a priori analysis extends the applicability of the
results from [52] to this setting with low regularity. The results are tested by an
extensive set of numerical experiments.

Acknowledgements
Studying the PhD is a time-consuming and involving project and during this busy
time, we certainly need help and support of many people. I am glad and happy to
acknowledge the support and help of many people for me during my PhD study
time at the University of Leicester. I would like to take this opportunity to express
my thanks and gratitude for all the people who helped me in my research, and I
will just name a few.
Firstly, I would like to express my indebtedness and gratitude to my supervi-
sors, Emmanuil Georgoulis and Andrea Cangiani, for their help, patience, support
and guidance during this long time. Their advice, encouragement and hints were
valuable and crucial in finishing my research.
Special thanks to my PhD colleagues and friends, Oliver, Sam, Zhaonan, Younis
and Stephen for their helpful discussions about the theory and implementation of
the finite element method and in particular in the finite element coding. Many
thanks to my friends in Michael Atiyah Building, Ali, Hassan, Mohammad, Omar,
Saeed, Mudher, Ahmed and Hoger for their help and encouragement. Also, I
would like to thank all the staff of the mathematics department for their help and
support. Lastly, all thanks and gratitude to all my darling family, my beloved
wife (Ehab), my kind parents, my dear sisters and my darling children (Ibraheem,
Yousif, and Mustafa), who without their support and encouragement I could not
finish my study. I thank them for their patience for the short time I spent with
them during this period.
iii

Contents
Acknowledgements iii
List of Figures vi
1 Introduction 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Contributions of this work and outline . . . . . . . . . . . . . . . . 8
2 hp-Version discontinuous Galerkin timestepping methods for parabolic
problems 11
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Useful inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Problem setup and the numerical method . . . . . . . . . . . . . . . 14
2.3.1 Preliminaries and the abstract setting . . . . . . . . . . . . . 14
2.3.2 Space–time Galerkin spaces . . . . . . . . . . . . . . . . . . 16
2.3.3 The fully discrete IMEX space–time ﬁnite element schemes . 17
2.4 Numerical implementation . . . . . . . . . . . . . . . . . . . . . . . 20
2.4.1 Discretisation in time . . . . . . . . . . . . . . . . . . . . . . 20
2.4.2 Discretisation in space . . . . . . . . . . . . . . . . . . . . . 23
2.5 hp–dG–timestepping for parabolic systems . . . . . . . . . . . . . . 25
2.6 Numerical examples and applications . . . . . . . . . . . . . . . . . 26
2.6.1 Example 1: Fisher system . . . . . . . . . . . . . . . . . . . 27
2.6.2 Example 2: Cycling Lotka–Volterra competition system . . . 29
2.6.3 Example 3: Predator–prey system . . . . . . . . . . . . . . . 31
3 A posteriori error analysis 33
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2 Reconstructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.2.1 Elliptic reconstruction . . . . . . . . . . . . . . . . . . . . . 36
3.2.2 Time reconstruction of ˜U . . . . . . . . . . . . . . . . . . . . 38
3.3 A posteriori error bounds . . . . . . . . . . . . . . . . . . . . . . . 41
3.3.1 Estimating the nonlinear term . . . . . . . . . . . . . . . . . 44
3.3.2 Completing the estimate . . . . . . . . . . . . . . . . . . . . 46
iv

CONTENTS v
3.3.3 Estimating the norms of σ and of . . . . . . . . . . . . . . 49
3.3.4 The ﬁnal a posteriori error bounds . . . . . . . . . . . . . . 53
3.4 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.4.1 Example 1: a linear problem . . . . . . . . . . . . . . . . . . 55
3.4.1.1 Example 1A: dG(1)–cG(2) scheme . . . . . . . . . 56
3.4.1.2 Example 1B: dG(2)–cG(2) scheme . . . . . . . . . 57
3.4.2 Example 2: a nonlinear problem . . . . . . . . . . . . . . . . 58
4 A priori error analysis 60
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.2 A priori error bounds . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.2.1 The stability of Pu − U . . . . . . . . . . . . . . . . . . . . 63
4.2.2 Completing the bound . . . . . . . . . . . . . . . . . . . . . 69
4.2.3 A priori error bounds . . . . . . . . . . . . . . . . . . . . . 73
4.3 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.3.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.3.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.3.3 Example 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.3.4 Example 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.3.5 Example 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5 Conclusions 85
5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
A Numerical computations of Chapter 2 88
A.1 Matrix form of the dG–timestepping schemes for semilinear parabolic
problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
A.2 Starting process on the previous time intervals . . . . . . . . . . . . 90
A.2.1 Starting process when  = 0 (The implicit case) . . . . . . . 91
A.2.2 Starting process when  = 1 (The implicit–explicit case) . . . 94
Bibliography 102

List of Figures
2.1 Example 1: Convergence history for dG(1)–cG(2) scheme for solv-
ing Fisher System. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2 Example 2: The solution at the final time T = 100: u1 in yellow,
u2 in blue, and u3 in red: (a) dG(1)–cG(1), (b) dG(1)–cG(2). . . . . 31
2.3 Example 3: The solution at the final time T = 163.46: (a) The Prey,
(b) The Predator, (c) The Prey and the Predator superimposed on
the same plot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.1 Example 1A. Convergence history for the dG(1)–cG(2) scheme with
kn = h (left) and kn = h3/2
(right). . . . . . . . . . . . . . . . . . . 57
3.2 Example 1B. Convergence history for the dG(2)–cG(2) scheme with
(right). . . . . . . . . . . . . . . . . . . 57
3.3 Example 2. Convergence history for the dG(1)–cG(2) scheme with
(right). . . . . . . . . . . . . . . . . . . 59
4.1 Example 1: h–version IMEX dG(r)–cG(2) scheme, r = 0, 1, 2, 3, 4,
for different error norms vs the time steps kn. . . . . . . . . . . . . 78
4.2 Example 2: h–version IMEX dG(r)–cG(4), r = 0, 1, 2, 3, 4 for dif-
ferent error norms vs the time steps kn. . . . . . . . . . . . . . . . . 80
4.3 Example 3: p–version IMEX dG timestepping scheme for r = 2 and
time step kn = 0.01, for different error norms. . . . . . . . . . . . . 81
4.4 Example 4: h–version IMEX dG timestepping dG(2)–cG(5) scheme
for different error norms. . . . . . . . . . . . . . . . . . . . . . . . . 83
4.5 Example 5: h–version on algebraically graded meshes dG(2)–cG(5)
for different error norms. . . . . . . . . . . . . . . . . . . . . . . . . 84
vi

To my dear wife Ehab, my loved parents, my beloved sisters,
and my lovely children, Ibraheem, Yousif and Mustafa.
vii

Chapter 1
Introduction
1.1 Background
The finite element method (FEM) is one of the most powerful, efficient, and general
techniques for solving partial differential equations, modelling a wide range of
problems in different areas such as biology, chemistry, physics, and engineering.
The FEM applies to a wide range of problems which can be written in variational
(weak) form and allows for high order approximations; its popularity and success is
especially due to its flexibility and accuracy in dealing with complicated problems
and geometries. The finite element method dates back to the 1940s in the works
of Hrennikoff and Courant, and builds their ideas and techniques on the works of
Galerkin, Rayleigh and Ritz. The method was then re–discovered in the 1950s by
engineers to solve common engineering problems, and after that has been studied
rigorously by mathematicians in the 1960s and 1970s. During long decades of
development, many engineers, scientists, and mathematicians have contributed to
the popularity of finite element methods, we refer to the following monographs for
more details [109, 95, 41].
In its earlier stages the finite element method started with standard continuous
finite element discretisations of the space variable following the Galerkin paradigm.
Typically, for the discretisation of the time variable, the conventional time stepping
methods are used such as Runge–Kutta or multistep methods. The finite element
1

Chapter 1 Introduction 2
in space and time is studied for the first time at the end of the 60s in the work of
Argyris and Scharpf [12]. The continuous Galerkin (cG) finite element method for
the time variable is first studied and analysed by Hulme [66, 65] in 1972 for ordinary
differential equations. Also, its relation to other collocation methods is considered.
Lesaint and Raviart [80] investigated the cG method for first order hyperbolic
neutron transport equations. The first detailed analysis for discontinuous Galerkin
(dG) time stepping schemes are carried out by Eriksson, Johnson and Thomée [47].
The directly–related discontinuous Galerkin finite element method for first order
hyperbolic problems is traced back to the work of Reed and Hill [93] in 1973.
Variational time–stepping methods nowadays are more popular, and they are gain-
ing increasing interest. Variational time–stepping methods of Galerkin–type are
based on weak formulations of the initial–value problems. They are known in
the literature by different names such as variational time discretisation methods,
variational time–marching schemes, variational time–advancing schemes and dG
or continuous Galerkin (cG) time–stepping schemes. For dG and cG schemes
the test spaces are discontinuous, i.e., they consist of discontinuous polynomials in
time, which naturally decouples the discrete Galerkin variational formulations into
local problems on each time step. In this work, we study discontinuous Galerkin
timestepping schemes; this is a family of arbitrary order timestepping methods
resulting in discontinuous, in general, approximations in the time variable. Dis-
continuous Galerkin timestepping methods can be also recast as certain families
of dissipative implicit Runge-Kutta methods upon suitable choices of quadrature
rules [80, 8]. In particular, dG timestepping schemes with quadrature at Gauss–
Radau points are equivalent to the implicit Runge–Kutta Radau method with
r intermediate stages (IRK–R(r)), where both are collocation methods. These
methods have attractive convergence properties in the discretisation of first order
derivatives mentioned above such as higher order convergence rates of order r + 1
for polynomial of order r and superconvergence of order 2r + 1 at the time nodes.
dG methods are convenient to use within adaptive algorithms whereby the time
and/or space meshes are adapted to the solution in an automatic fashion, typically
driven by a posteriori error estimators; this is due to the lack of necessity of any
continuity requirements between timesteps which can allow for locally variable

order approximations and local timestepping. Also, dG timestepping schemes of
order r are equivalent at the nodal points with the standard difference subdiagonal
Padé schemes of order (r, r − 1) [109].
Moreover, classical time-stepping methods are not appropriate for problems with
time-dependent domains (variable domains) or time-dependent free boundary prob-
lems. For the treatment of such problems, the use of variational space-time meth-
ods is essential [24]. Recent works [25, 23, 22, 24] have examined higher order
time discrete arbitrary Lagrangian Eulerian (ALE) formulation by the use of dG
time-stepping schemes. The authors performed both the a posteriori and a pri-
ori error analysis as well as the stability analysis for higher order discontinu-
ous Galerkin time stepping schemes for ALE problems. Also, the discontinuous
Galerkin time variational schemes played an important role in the study of op-
timal control problems. Chrysafinos and coworkers [32, 35, 34, 33] studied the
convergence of optimal control problems related to semilinear parabolic equations
such as FitzHugh-Nagumo system and evolutionary Stokes equations associated
with constrained optimal control problems by using discontinuous time stepping
methods of arbitrary orders using the continuous finite element method in space.
Sudirham and coworkers [108] examined space–time Galerkin discretisation for the
advection–diffusion problems in the context of the ALE formulation.
The increasing popularity of a posteriori error estimates in deriving efficient and
accurate adaptive methods that reduce the cost and time of computations has put
forth the need to develop such estimates for numerical methods for more com-
plex/nonlinear problems. Indeed, in recent years, adaptive finite element methods
have become important tools in solving complicated problems such as problems
with local singularities such as singularities arising from sharp shock–like fronts,
interior or boundary layers, and re-entrant corners, and they are the subject of in-
tensive research and study. A posteriori error analysis for stationary/elliptic prob-
lems has been studied widely, and important developments have been achieved. On
the other hand, the study of nonlinear stationary problems and time–dependent
problems is still not yet mature [111, 3]. In particular, the study of time hp–
adaptivity, space-time–hp–adaptivity (fully hp–adaptivity), and a posteriori error

analysis for the Galerkin variational time–stepping methods, used in the context
of numerical solution of nonlinear evolution partial differential equations, has not
been addressed before in the literature.
1.2 Literature review
Space-time variational methods for the discretisation of evolutionary PDEs are
becoming more popular as evidenced by the recent and ongoing research in this
field [107, 105, 26]. Error analysis and aspects of implementation of these methods
have been studied by many authors [68, 37, 80, 13, 63, 64, 47, 52, 82, 79, 2, 85,
67, 74, 94, 99, 103, 100, 101, 6, 5, 21, 97, 73, 17, 114, 106, 25, 23, 22, 24, 62, 118].
In [37], the authors investigated the application of dG timestepping schemes for
linear non–stiff ordinary differential equations. The first error analysis for linear
parabolic problems is done in [68] by Jamet. In [97] Schieweck studied the sta-
bility properties of cG timestepping schemes. The authors in [62, 118] presented
a novel unified framework of discontinuous Galerkin method for deriving different
time stepping schemes, via different boundary conditions for the time variable,
numerical quadrature and test functions. Aziz and Monk [13] investigated the cG
method and they showed that the cG(1), i.e., continuous Galerkin methods with
linear elements is equivalent to the Crank–Nicolson method with time averaged
data. Also, they derived error estimates for this method. These methods have
received considerable interest in the context of space–time adaptivity throughout
the years, as they offer a variational, arbitrary order timestepping framework and,
crucially, allow for locally variable timestep sizes in different spatial regions of the
computational domain. On the other hand, one of the most challenging aspects
of implementation of variational time stepping schemes "such as discontinuous
Galerkin timestepping methods" is the high computational cost (memory size and
implementation time) for solving the block algebraic linear systems arising from
using these methods for discretising time-dependent PDEs or ODEs. The reduc-
tion of the computational cost of the implementation of variational discontinuous
or continuous time marching schemes is one of the challenging issues in using these

methods. Many researchers have considered these issues from different numerical
view points such as [99, 114, 94, 106]. Also, in [114] Basting and Weller proposed
and analysed efficient preconditioners for block algebraic linear systems result-
ing from solving linear parabolic equations by variational time stepping methods.
Richter, Springer and Vexler [94] analysed the solution of the nonlinear systems
arising from solving nonlinear parabolic equations by using discontinuous Galerkin
methods of order r in time. They avoided the inevitable complex coefficients aris-
ing from direct decoupling of the nonlinear systems by a judicious use of the
Newton method. Also, in the recent article [106], Smears derived a fully robust
and efficient preconditioning scheme for the block algebraic linear systems aris-
ing from the solution of parabolic problems by dG variational time discretisation
schemes.
All these works employed the h–version dG timestepping schemes where the ap-
proximation order r is fixed and usually low order, while decreasing the time steps.
This approach leads to algebraic rates of convergence of order r + 1 for smooth
solutions in time. The p– and hp–versions of the FEM were initiated in the 1980s
by Babˇuska, Szabö and their co-workers [14, 15? ]
The p– and hp–versions Galerkin timestepping methods can solve the transient
problems which have smooth solutions with local singularities with high algebraic
and even exponential convergence rates, and their analysis have been subject of
great interest, see e.g. [99, 100, 101, 103, 102, 115, 60, 76, 90, 98, 117]. In par-
ticular, Schötzau and Schwab initiated and introduced the hp–version Galerkin
time-stepping methods in a series of papers [103, 100, 101, 102], where they stud-
ied, analysed and examined hp–dG–time stepping methods for the initial value
ODE problem to the fully discrete canonical parabolic problem, proving new ex-
plicit a priori error estimates for the approximations orders and time steps and
showing that dG time stepping methods have exponential/spectral accuracy for
smooth time–dependent problems. Mustapha [90] examined the numerical solution
of the fractional subdiffusion problems by the use of hp–time stepping discontinu-
ous Galerkin methods. The solution of nonlinear PDEs by hp–dG time–advancing
schemes have attracted more research and interest recently. In [105] and [107] the

authors studied the numerical solution of the nonlinear Hamilton–Jacobi–Bellman
equation by using fully discrete hp– and hp−τq– versions of discontinuous Galerkin
time stepping methods respectively. Janssen and Wihler in [69] investigated hp–
Galerkin time stepping methods for nonlinear initial value problems. They proved
the existence results for the continuous and discontinuous Galerkin methods for
problems with Lipschitz–type nonlinearities and blow–up in finite time. We also
note the recent work [76] on adaptive hp-version dG-timestepping methods for
finite time blow–up detection in semilinear parabolic problems.
Rigorous a posteriori error bounds for numerical methods for evolution prob-
lems are now a mature yet significantly expanding subject. The a posteriori
error analysis of standard numerical methods for linear parabolic problems has
been studied by many researchers. The classical works for the a posteriori er-
ror analysis for the dG timestepping schemes started with the seminal works of
Erkisson, Johnson et al. [42, 43, 44, 45], in which they were studied and anal-
ysed space–time finite element methods involving dG–timestepping via duality
techniques; see also [51, 53]. Picasso [91] showed a posteriori bounds of residual
type for backward Euler timestepping methods. The idea of reconstruction was
introduced in 2003 by Makridakis and Nochetto [83] for deriving optimal order
a posteriori error estimates for semi-discrete linear parabolic problems through
the energy method, and was further developed for the case of fully–discrete linear
parabolic problems in [77]. A significant body of literature following in this vein is
[7, 8, 83, 38, 59, 78, 50, 18, 74, 87, 92, 88]; we also refer to the survey [81] in which
a general overview and treatment for the reconstruction technique is given. We
note that the dG–timestepping reconstruction from [84] utilises the Gauss-Radau
nodes, which are known to be points of superconvergence for the dG method in one
dimension; see also [96] for a review of superconvergence in dG methods and the
related question of postprocessing. A posteriori error analysis for linear parabolic
problems for space-time methods has also received renewed attention during the
10 years or so: there has been a renewed interest in the derivation of rigorous a
posteriori error bounds for dG timestepping schemes [84, 98, 76, 48, 49, 60, 55].
In spite of the progress made in the a posteriori error analysis of linear parabolic

problems discretised by traditional and classical time-marching schemes, semi-
linear and, generally, nonlinear evolution equations pose a number of additional
challenges. These include the treatment of nonlinear reactions, the proof of lower
bounds, etc. An interesting approach for semilinear parabolic problems is the use
of so-called continuation arguments for the proof of a posteriori bounds. This
idea appeared in [72] and further developed in [19, 20] for the Allen-Cahn and
the Ginzburg-Landau equations and related phase–ﬁeld models; see also [61, 54].
Related to this, in [75], the authors studied and derived the error estimates for
blow-up solutions for semilinear parabolic equations which was further developed
in the fully-discrete setting in [30]. All these developments in using continua-
tion arguments for the proof of a posteriori bounds assumed standard low order
timestepping schemes and, in particular, the backward Euler method.
Therefore, the proof of a posteriori error bounds for arbitrary order space–time
methods involving dG-timestepping for nonlinear parabolic problems with strong
nonlinearities (e.g., non globally Lipschitz) remains an interesting challenge which
we aim to address in this thesis. At the same time, the proof of standard a priori
error bounds for the same family of methods under such weak assumptions on the
nonlinear growth is also elusive.
The a priori error analysis for classical timestepping schemes is now understood
at large, see e.g. [116, 110, 28, 20, 29] and the references therein. In [28, 29],
the authors examined the a priori error analysis for semilinear interface parabolic
problems. The variational time–marching schemes are considered by many authors
in diﬀerent contexts [47, 52, 82, 101, 113, 117, 109, 23, 26]. Schötazu and Schwab
[101] analysed and derived the a priori error estimates for the hp-version dG
timestepping methods for initial value problems. Wihler [117] investigated the
a priori error bounds for the hp–version cG timestepping schemes for nonlinear
initial value problems.

1.3 Contributions of this work and outline
In this work, we consider the numerical solution of semilinear parabolic problems
by using discontinuous Galerkin timestepping schemes in conjunction with con-
forming finite elements in space. To avoid the necessity of solving a nonlinear sys-
tem for each time step, an implicit–explicit dG timestepping scheme is employed.
This approach was first introduced in [52]. For this method we prove a posteriori
and a priori error bounds for the case of merely locally Lipschitz nonlinear reac-
tions that are not assumed to satisfy any monotonicity properties, thereby there
is no coercivity in a stronger norm than the standard L∞(L2) + L2(H1
)–norm.
We are firstly concerned with the derivation of hp–version a posteriori error bounds
in the L∞(L2)– and L2(H1
)–norms for fully discrete implicit–explicit (IMEX)
methods of variable order for semilinear parabolic problems of reaction-diffusion
type. The nonlinear reaction term is assumed to be locally Lipschitz and satisfy-
ing a growth condition in the spirit of [110]. The time discretisation consists of a
hp–version discontinuous Galerkin method treating implicitly the diffusion spatial
operator, and using an explicit multistep method for the nonlinear reaction term.
This is combined with the standard conforming finite element method used for the
spatial discretisation. The multistep IMEX–dG time discretisation we consider in
this work was introduced in [52], whereby a priori error bounds were proven for
the case of globally Lipschitz nonlinear reactions. To reduce the computational
overhead, the nonlinear reactions are treated explicitly via sufficiently high–order
interpolation of solution values from previous timesteps [52]. Therefore, the so-
lution of one linear system per timestep is required. The proof combines the
recent space–time reconstruction proposed in [60], along with a suitable implicit
perturbation of the explicitly discretised nonlinear reaction part in the spirit of
[58, 57]. The treatment of the non-Lipschitz nonlinearity involves a continuation
argument in the spirit of [19, 28, 30] along with suitable Sobolev imbeddings. To
the best of our knowledge, this is the first time such a posteriori error bounds
for the fully–discrete methods involving dG–timestepping for nonlinear evolution
PDEs appeared in the literature. Crucially, no a priori Courant-Friedrichs-Lewy
(CFL) type conditions (with the respective often obscure constants involved) will

be required for the validity of our a posteriori error bounds for explicit timestep-
ping methods (cf., also [58, 57]). Indeed, for unstable combinations of local spatial
and temporal meshsizes, the a posteriori estimator remains reliable. In fact, this
remarkable property motivates the study of a posteriori estimation of CFL con-
stants as a non-standard potential use of rigorous a posteriori error upper bounds
for (implicit–)explicit methods; this will be discussed elsewhere, as it is beyond
the scope of this work.
On the other hand, the a priori error analysis for dG–timestepping schemes is both
classical [47, 52, 109] and modern [36, 25, 113, 69, 70] in that a number of issues
regarding regularity assumptions of the exact solution and the treatment of chal-
lenging nonlinearities have received considerable interest lately. In this work, we
derive hp–version a priori error bounds for the fully–discrete IMEX–dG timestep-
ping scheme discussed above in L∞(L2) and L2(H1
) norms. These error bounds are
also explicit with respect to the local, possibly varying, order in the time discreti-
sation (hp-version a priori error estimates). They are derived via a combination of
classical ideas, such as the use of an elliptic projection technique and the discrete
Grönwall inequality. Also, we employed enhanced discrete stability estimates in
the H1
(L2)–seminorm. We applied ideas from the recent works [27, 26] and ex-
tending them to the IMEX-dG discretisation of semilinear parabolic problems with
merely locally-Lipschitz continuous nonlinear reaction term, thereby generalising
the applicability of the results from [52] with lowest possible regularity. Also,
we derived these error bounds with signiﬁcantly less restrictive assumptions on
the nonlinear reaction growth. Moreover, to the best of our knowledge, there are
no previous results on a priori error bounds for fully–discrete methods involving
dG–timestepping for nonlinear evolution PDEs, with locally-Lipschitz continuous
nonlinearities.
The remainder of this thesis is organised as follows. Chapter 2 is introductory:
we introduce some notation, and deﬁne the space–time scheme and space–time
reconstruction operators, and derive the fully discrete implicit–explicit (IMEX)
method of variable order for semilinear parabolic problems. Furthermore, a series
of numerical examples investigating the performance of the numerical method

for solving semilinear parabolic systems from biology and ecology are given. In
Chapter 3, we consider the derivation of a posteriori error bounds for the fully
discrete semilinear parabolic problems in L∞(L2) and L2(H1
) norms. Also, a set
of numerical examples for linear and nonlinear parabolic equation highlighting
the performance of the a posteriori error estimates are presented. A priori error
bounds for the fully–discrete semilinear parabolic problems in L∞(L2) and L2(H1
)
norms with as set of numerical experiments testing the validity of these error
bounds are presented in Chapter 4. Some conclusions are given in Chapter 5.

Chapter 2
hp-Version discontinuous
Galerkin timestepping methods
for parabolic problems
2.1 Introduction
Discontinuous Galerkin timestepping methods are arbitrary order single step im-
plicit dissipative methods. Due to this, they are suitable for dissipative evolution
equations and, in particular, for various classes of parabolic problems. The order
of convergence of dG(r) time–marching methods of polynomial degree r is r+1 for
sufficiently smooth exact solutions. Also, these methods are A–stable and some-
times strongly A–stable, for more details see [97]. Another appealing feature of the
discontinuous Galerkin timestepping methods, is that they require lower regularity
of solutions compared to other timestepping schemes, and they naturally allow for
locally variable time steps (i.e., different timestep sizes at different parts of the
spatial domain) and variable polynomials orders making them more convenient
for h–, p–, and hp–versions time-stepping schemes. Consequently, they are more
relevant to use for h–, p–, hp–adaptivity.
11

Chapter 2 hp-Version discontinuous Galerkin timestepping methods for parabolic
problems 12
Here we study a fully–discrete implicit time–discontinuous and spatially conform-
ing Galerkin scheme for evolutionary semilinear parabolic problems. For conve-
nience in this chapter, we will follow Rothe’s approach by firstly discretising in
time, and then discretising the resulting scheme in space to obtain at the end the
fully discrete space–time scheme for parabolic problems. Finally, we will present
a series of numerical applications of dG–timestepping schemes of various orders in
mathematical biology and mathematical ecology.
Remark 2.1. Throughout the thesis, the constant C is used to denote an arbitrary
real constant, and it is not necessarily the same each time it occurs.
2.2 Useful inequalities
In this section, we recall from [95] some inequalities which will be used frequently
in the remaining of this thesis.
Definition 2.2 (Hölder’s inequality). Let 1 ≤ p, q ≤ ∞ such that 1
p
+ 1
q
, then for
any u ∈ Lp(Ω) and v ∈ Lq(Ω), the product uv ∈ L1(Ω), and we have
|(u, v)Ω| ≤ u Lp(Ω) v Lq(Ω). (2.1)
"Note that" the Cauchy–Schwarz’s inequality is a special case of the Hölder’s in-
equality when p = q = 2.
Definition 2.3 (Young’s inequality). For every a, b ∈ R, and for every ε > 0, we
have
ab ≤
ε
2
a2
+
1
2ε
b2
. (2.2)
Definition 2.4 (Continuous Gronwall’s inequality). Let u, v, w be piecewise con-
tinuous nonnegative functions defined on the interval (a, b). Assume that v is
nondecreasing function and that there is a positive constant C independent of t

problems 13
such that, for all t ∈ (a, b),
u(t) + w(t) ≤ v(t) + C
t
a
u(τ) dτ. (2.3)
Then, for all t ∈ (a, b),
u(t) + w(t) ≤ eC(t−a)
v(t). (2.4)
Definition 2.5 (Discrete Gronwall’s inequality). Let (an)n, (bn)n, and (cn)n be
sequences of nonnegative numbers satisfying, for all n ≥ 0,
an ≤ bn +
n
i=0
ciai. (2.5)
Then, for all n ≥ 0,
an ≤ bn +
n
i=0
bici exp
n
j=i
cj . (2.6)
Definition 2.6 (Poincaré–Friedrichs inequality). There is a positive constant C
such that for every v ∈ H1
(Ω),
v L2(Ω) ≤ C v L2(Ω) + |
∂Ω
v| . (2.7)
As special case, we have, for every v ∈ H1
0 (Ω),
v L2(Ω) ≤ C v L2(Ω). (2.8)
Definition 2.7 (Sobolev imbedding inequality). For 1 ≤ q < ∞, and if Ω ⊂ R2
,
the space H1
0 (Ω) is imbedded into the space Lq(Ω), i.e., for every v ∈ H1
0 (Ω),
v Lq(Ω) ≤ C v L2(Ω). (2.9)

problems 14
2.3 Problem setup and the numerical method
2.3.1 Preliminaries and the abstract setting
For H a real Hilbert space and I = [a, b] ⊂ R, the Bochner space Lp(I; H) is
defined by Lp(I; H) := {v : I → H such that v Lp(I;H) < ∞}, with the respective
norm given by
v Lp(I;H) :=



I
v(t) p
H dt
1/p
, for 1 ≤ p < ∞,
ess sup
t∈I
v(t) H, for p = ∞.
Upon denoting by v the (weak) derivative of v with respect to the “time”-variable
t ∈ I, we can also define the Sobolev-Bochner spaces of order k (with respect to
the time derivatives), where k is a positive integer, as
Wk
p (I; H) := {v, v , v , · · · , vk
: I → H such that v Wk
p (I;H) < ∞}.
As special case, we define
W1
p (I; H) := {v, v : I → H such that v W1
p (I;H) < ∞},
and v W1
p (I;H) := v p
Lp(I;H)+ v p
Lp(I;H)
1/p
. When H, (·, ·)H is a Hilbert space
with respective inner product, L2(I; H) and H1
(I; H) ≡ W1
2 (I; H) are also Hilbert
spaces endowed with the inner products I(v(t), w(t))H dt and I(v(t), w(t))H +
(v (t), w (t))H dt, respectively. We may also write Z(a, b; H) instead of Z(I; H) for
Z ∈ {Lp, W1
p }, see [109, 95].
We also denote by C(I; V) := {v : I −→ V : v is continuous} the space of
continuous in time functions equipped with the norm
v C(I;V) := sup
t∈I
v(t) V.

problems 15
Let V ⊂ H be another Hilbert space with norm · V and let V∗
denote its dual
space defined on the space of all functions z for which the norm
z V∗ := sup
0=v∈V
(z, v)H
v V
,
is finite. The spaces V, H and V∗
form a, so-called, Gelfand triple
V → H → V∗
,
with the duality pairing ·, · V∗×V extending the inner product (·, ·)H, in the sense
that, for all u ∈ H and v ∈ V holds u, v V∗×V = (u, v)H. The subscript V∗
× V
in the duality pairing will be omitted whenever no confusion is likely to occur.
Although we shall work within the above abstract setting, a typical case is when
H = L2(Ω), V = H1
0 (Ω), V∗
= H−1
(Ω).
We consider the semilinear parabolic initial value problem:
find u ∈ H1
(0, T; V∗
) ∩ L2(0, T; V) such that
u + Au = f(·, u) for all t ∈ I, u(0) = u0, (2.10)
for some known function u0 ∈ H, u = u(t, x), x ∈ Rd
, where d is a positive
constant, and A : V −→ V∗
is a linear elliptic operator, which is continuous and
coercive(elliptic) with respect to the norm of V. We also define the bilinear form
a : V × V −→ R associated with A by
Av, w V∗×V = a(v, w) for all v, w ∈ V, (2.11)
which inherits the continuity and coercivity properties of A, viz.,
|a(v, w)| ≤ Ccont v V w V for all v, w ∈ V, (2.12)
a(v, v) ≥ Ccoer v 2
V for all v ∈ V, (2.13)
with Ccont, Ccoer positive constants independent of v, w. Of course, the analysis
presented below can be generalised to the case where a satisfies a Gärding-type

problems 16
inequality instead; we prefer to keep the presentation simple by assuming just he
coercivity of a instead throughout.
The function f : I × Rd
× R → R is smooth and locally Lipschitz–continuous,
bounded in the first two arguments and satisfies the growth condition for the
third argument [110]:
|f(t, x, z1) − f(t, x, z2)| ≤ C|z1 − z2|(1 + |z1| + |z2|)r
, for r ≥ 0, (2.14)
for all z1, z2 ∈ R with | · | denoting the Euclidean distance. Here, C is a positive
constant, uniform with respect to the first two arguments. The range of r will be
further constrained from above in what follows, depending on the dimension of
the spatial computational domain Ω ⊂ Rd
. In what follows, we shall often sup-
press for brevity the dependence of f on its first two arguments writing, therefore,
f(t, x, w) = f(w). Generalisations of the above assumptions in the first two ar-
guments are possible in the context of certain Caratheodory-type conditions, but
we refrain from discussing these in the interest of simplicity of the presentation.
Crucially, however, we do not assume any monotonicity of the nonlinear reaction
terms. As a result, we do not have any extra control in norms other than the
respective linear problem.
2.3.2 Space–time Galerkin spaces
Let I = [0, T] be the time interval with final time T > 0 and, for 0 = t0 <
t1 < ... < tN = T, consider the partition {In, n = 0, ..., N} of I into subintervals
In := (tn−1, tn] for n = 1, ..., N, and I0 := {0}, with corresponding timesteps
kn := tn − tn−1, n = 1, 2, ..., N. We also consider a finite sequence {Vn}N
n=0 with
Vn ⊂ V, n = 0, ..., N of conforming finite element subspaces of V, associated with
the time subintervals In.
Let H be a Hilbert space. We define
Pr
(R; H) := {p : R → H : p(t) =
r
i=0
ψiti
, ψi ∈ H, i = 0, 1 . . . , r},

problems 17
as the space of H–valued polynomials of degree at most r. More generally, we
define Pr
(I; H) := {p|I : p ∈ Pr
(R; H)}. We also consider the time-discrete and
the space-time finite element subspaces
Yn(S) := Prn
(In; S), S ∈ {H, V}, and Xn( ˜S) := Prn
(In; ˜S), ˜S ∈ {Vn, Vh},
respectively, for all n = 0, 1, . . . , N with rn denoting the local temporal polyno-
mial degree, which may vary from one timestep to another, and Vh ⊂ V is a
(conforming) finite element space.
We can then define the time–discrete and the space–time Galerkin spaces
Y(S) ≡ Yr(S) := {v : [0, T] → S : v|In ∈ Yn(S), n = 0, 1, . . . , N},
and
X( ˜S) ≡ Xr( ˜S) := {v : [0, T] → ˜S : v|In ∈ Xn( ˜S), n = 0, 1, . . . , N},
respectively, often suppressing the dependence on the polynomial degree vector
r := (r1, r2, . . . , rN ) for brevity.
Moreover, for a piecewise continuous function v : I ⊂ R → H, with the time nodes
tn as possible points of discontinuity, we define the time–jump
[v]n := v+
n − v−
n ,
where v±
n := limδ→0+ v(tn ± δ), the respective one–sided (right and left) limits for
n = 0, 1, . . . , N. For more details, see [99, 73].
2.3.3 The fully discrete IMEX space–time finite element
schemes
In this section, we drive the fully–discrete implicit time discontinuous and spa-
tially continuous Galerkin discretisation of the problem (2.10). Writing the model

problems 18
problem (2.10) in weak form by testing it with a smooth function χ and integrat-
ing over the spatial domain Ω, and also integrating in time over the time domain
I = [0, T], we have
T
0
(u , χ)H + a(u, χ) dt =
T
0
(f(u), χ)H dt, ∀χ ∈ X. (2.15)
Now, integrating by parts the first term in (2.15) and letting χ(T) = 0, we obtain
T
0
− (u, χ )H + a(u, χ) dt = (u0, χ(0))H +
T
0
(f(u), χ)H dt ∀χ ∈ X. (2.16)
Now, approximating the exact solution u by a function U ∈ Xn, yields
T
0
− (U, v )H + a(U, v) dt = (U−
0 , v+
0 )H +
T
0
(f(u), v)H dt ∀v ∈ Xn. (2.17)
Integrating by parts the first term in (2.17) in each time interval In, and noting
that v+
T = 0, implies
−
T
0
(U, v )H dt = −
N
n=1
(U, v)H|tn
tn−1
−
tn
tn−1
(U , v) dt ∀v ∈ Xn
=
T
0
(U , v)H dt +
N
n=2
([U]n−1, v+
n−1)H + (U+
0 , v+
0 )H. (2.18)
Substituting (2.18) in (2.17), we arrive at
T
0
(U , v)H + a(U, v) dt +
N
n=2
([U]n−1, v+
n−1)H + (U+
0 , v+
0 )H
= (U−
0 , v+
0 )H +
T
0
(f(u), v)H dt ∀v ∈ Xn. (2.19)
Due to the discontinuity of v ∈ Xn, choosing v = 0 outside the time interval In
decouples the problem (2.19) into one problem on each time interval In for n ≤ N.
Finally, we arrive at the fully–discrete implicit time discontinuous and spatially
conforming Galerkin approximation of (2.10) which reads: set U−
0 := ˜P0u0 and
find U ∈ X such that
In
((U , v)H + a(U, v)) dt + ([U]n−1, v+
n−1)H =
In
(f(U), v)H dt (2.20)

problems 19
for all v ∈ Xn and for n = 1, ..., N, Here ˜P0 denotes the elliptic projection operator
and will be defined later, and [U]n = U+
n − U−
n , see [109]. The space-time method
(2.20) is fully implicit in the sense that a nonlinear system of equations for the
numerical degrees of freedom has to be solved at each time interval.
Aiming for a linearly implicit method, we follow [52] and we replace f(U) in (2.20)
by its linear interpolant in time Πf(U), defined so that Πf(U)|In ∈ Pµ
(In; Vn),
for all n = 1, . . . , N, where µ = 2rn, using values of U from previous time intervals
Im, m < n only and extrapolating the resulting interpolant into In. In this case,
the solution process will result in a linear system for U per time-step, giving rise
to an implicit–explicit (IMEX) method. Of course, one can also interpolate on
the previous and the current time intervals Im, m ≤ n. This case will lead to
a nonlinear system of equations for U, although it can potentially be easier to
implement for certain nonlinearities f. In both cases, the time interpolant Πf(U)
can be represented on each In as
Πf(U)(t)|In := Πµ
n−f(U)(t) :=
n−
η=n−−µ
ξη(t)f(tη, ·, U−
η ), (2.21)
where Πµ
n−,  = 0, 1, is the interpolation operator for polynomials of degree µ
at the nodes tn−−µ, . . . , tn− and ξη are the respective Lagrange basis functions
defined as follows:
ξη = Πn−
i=n−−µ
(t − ti)
(tη − ti)
, i = η,  = 0, 1, (2.22)
for η = n −  − µ, · · · , n − . The corresponding IMEX space–time scheme reads:
set U−
0 := ˜P0u0 and find U ∈ X such that
In
((U , v)H + a(U, v)) dt + ([U]n−1, v+
n−1)H =
In
(Πf(U), v)H dt (2.23)
for all v ∈ Xn, for n = µ + , ..., N. Of course, as this is a multistep method,
we can only use it after a certain number of time-steps, depending on the order
of the method. Without loss of convergence rate, however, we can consider a few
(very small in size) timesteps with the zeroth order method, i.e., the implicit Euler
method with explicit treatment of the nonlinear reaction, before using (2.23) with

problems 20
higher order than zero. The interpolant degree µ = 2rn is required to represent
the integrand (Πf(U), v)H without loss of convergence rates. Finally, for  = 1,
we arrive at the IMEX method, while, for  = 0, we retrieve the fully implicit
scheme; for further details we refer to [52]. Note that the values U−
η are known to
be points of superconvergence for the respective time-discrete problem, where the
method has superconvergence rates 2r + 1 at these points [71, 6]; see also [96].
2.4 Numerical implementation
In this section, we derive the numerical implementation of (2.23) by introducing
appropriate basis for the space–time trial and test spaces, to arrive at a formulation
where the numerical scheme can be computed by iterating through the time steps
(time slabs) and solving a linear system at every time step. Firstly, we discretise
in time and subsequently we discretise the resulting problem in space.
2.4.1 Discretisation in time
The discrete solution U is a polynomial function of the time variable of degree r.
As such, it can be written in terms of basis functions φn,j(t) ∈ Yn, ∀j = 0, 1, ..., r
as
U(t) :=
r
j=0
Uj
nφn,j(t), ∀t ∈ In, (2.24)
and
U (t) :=
r
j=0
Uj
nφn,j(t), ∀t ∈ In, (2.25)
where: Uj
n are the coeﬃcients in the ansatzes (2.24) and (2.25) are elements of the
space Vn and the basis functions φn,j are Lagrange polynomials of degree r deﬁned
for r + 1 nodal (support) points tn,j ∈ In, j = 0, 1, ..., r, to satisfy the conditions
φn,j(ti) = δi,j, ∀i, j = 0, 1, ..., r, (2.26)

problems 21
where δi,j is the Kronecker delta
δi,j =



1 if i = j,
0 if i = j.
(2.27)
The test functions in (2.23) can be chosen as products of two functions v =
vφn,i, i = 0, 1, ..., r where v ∈ Vn (space function) and φn,i ∈ Xn (time function).
Now, setting v = vφn,i and using (2.21), we can rewrite (2.23) as an In–problem:
set U−
0 := ˜P0u0 and find U ∈ X such that for all v ∈ Vn it holds
In
(U , v)H + a(U, v) φn,i dt + ([U]n−1, v)Hφn,i(tn−1)
=
In
n−
η=n−−µ
ξη(f(U−
η ), v)Hφn,i dt
(2.28)
for i = 0, 1, ..., r. We note this interpolant can be applied from the (µ + 1)th
interval i.e on Iµ+1, · · · , IN , but can not be used on the first (µ) intervals i.e. on
I1, · · · , Iµ. For this reason we need to construct special interpolants for the first
µ intervals. In the particular case when the solution is a constant polynomial, i.e.
when rn = 0, then the interpolant also is a constant polynomial, and, obviously,
the constant interpolant can be applied for all the intervals I1, · · · , In.
By inserting the representations (2.24) and (2.25) in (2.28) we have the following
linear algebraic system for the r + 1 unknown coefficients Uj
n ∈ Vn, j = 0, 1, ..., r:
r
j=0
(Uj
n, v)H
In
φn,jφn,i dt +
r
j=0
a(Uj
n, v)
In
φn,jφn,i dt
+
r
j=0
(Uj
n, v)Hφn,j(tn−1)φn,i(tn−1)
= (U
(0)
n−1, v)Hφn,i(tn−1) +
n−
η=n−−µ
(f(U−
η ), v)H
In
ξηφn,i dt,
(2.29)
where U
(0)
n−1 = U−
n−1 is the initial condition on the time interval In and, hence, it
is obtained from the solution on the previous time interval In−1.
The time integrals over In in (2.29) with the basis functions, test functions and

problems 22
support points are mapped into the reference interval Î = [0, 1] and all computa-
tions are subsequently performed on the reference interval Î. For this reason we
define the affine domain transformation Tn : Î −→ In, such that
t := Tn(ˆt) = kn
ˆt + tn−1, ∀ˆt ∈ Î, n = 1, ..., N, (2.30)
and the inverse reference mapping T−1
n : In −→ Î that maps back from the
reference interval Î to the domain interval In, given by
ˆt := T−1
n (t) = (t − tn−1)/kn, ∀t ∈ In, n = 1, ..., N. (2.31)
We define the reference basis functions and reference test functions ˆφj ∈ Pr
(Î; V), j =
0, 1, ..., r on the reference interval Î to be Lagrange polynomials of order r ≥ 0
with respect to r + 1 nodal points ˆtj ∈ Î, j = 0, 1, ..., r such that
ˆφj(ˆti) = δi,j, ∀i, j = 0, 1, ..., r. (2.32)
The corresponding support points in the original domain interval In are given by
tn,j = Tn(ˆtj), j = 0, 1, ..., N.
Similarly, the relation between the original basis and the reference basis is
φn,j(t) := φn,j(Tn(ˆt)) = φn,j(t) ◦ Tn(ˆt) = ˆφj(ˆt), ˆt ∈ Î, ∀n = 1, ..., N. (2.33)
Now, by mapping the time integrals in (2.29) to the reference interval Î we obtain
r
j=0
(Uj
n, v)
Î
ˆφj
ˆφi dˆt + kn
r
j=0
a(Uj
n, v)
Î
ˆφj
ˆφi dˆt +
r
j=0
(Uj
n, v)ˆφj(0)ˆφi(0)
= (U
(0)
n−1, v)ˆφi(0) + kn
n−
η=n−−µ
(f(U−
η ), v)
Î
ˆξη
ˆφi dˆt.
(2.34)
For brevity, we denote the integrals and coefficients in (2.34) by
αi,j :=
Î
ˆφj
ˆφi dˆt, i, j = 0, 1, ..., r, (2.35)

problems 23
βi,j :=
Î
ˆφj
ˆφi dˆt, i, j = 0, 1, ..., r, (2.36)
γi,j := ˆφj(0)ˆφi(0), i, j = 0, 1, ..., r,
σi := ˆφi(0), i = 0, 1, ..., r,
and
i,η :=
Î
ˆξη
ˆφi dˆt, η = n −  − µ, · · · , n − , i = 0, 1, · · · , r,  = 0, 1,
thereby arriving at
r
j=0
i,j(Uj
n, v) + knβi,ja(Uj
n, v)
= σi(U
(0)
n−1, v) + kn
n−
η=n−−µ
i,η(f(U−
η ), v), i = 0, 1, ..., r,
(2.37)
where
i,j := αi,j + γi,j, i, j = 0, 1, ..., r. (2.38)
2.4.2 Discretisation in space
In this section, we expand upon the spatial discretisation of (2.37) in view of
deriving the complete space–time discrete schemes. Since the discrete functions
Uj
n belong to the discrete space Vn they can be written as a linear combination of
its basis functions. Let nh be the dimension of Vn, the number of the degrees of
freedom (dofs) in space at each time step. Assume that a set of nodal dofs is given
and let ζl(x) ∈ Vn be the corresponding Lagrangian basis. Let Uj
n ∈ Rnh be the
vector of nodal values associated to the functions Uj
n ∈ Vn. Then, the approximate
finite element solution Uh(t, x) is written as
Uj
n(t, x) :=
nh
l=1
(Uj
n)l ζl(x) ∀t ∈ In, x ∈ Ω. (2.39)
The approximate time discrete solution U(t) ∈ V in (2.24) is approximated now by
space–time fully discrete finite element solution Uh(t, x) ∈ Vn and is represented

problems 24
on the reference time interval Î by
Uh(ˆt, x) :=
nh
l=1
r
j=0
(Uj
n)l ζl(x)ˆφj(ˆt) ∀ˆt ∈ Î, x ∈ Ω. (2.40)
The nonlinear function f(U), and, consequently, the nonlinear term (f(U−
η ), v)
reads
f(U−
η ) :=
nh
l=1
f(U−
η )l ζl(x) ∀x ∈ Ω, (2.41)
(f(U−
η ), v)H = (
nh
l=1
f(U−
η )l ζl, ζs)H =
nh
l=1
f(U−
η )l(ζl, ζs)H, (2.42)
where the subscript l represents the values of the interpolant and the basis func-
tions at the nodal points.
Hence, after inserting all these terms into (2.37), we get
r
j=0
i,j
nh
l=1
(Uj
n)l(ζl, ζs)H + kn
r
j=0
βi,j(Uj
n)l
nh
l=1
a(ζl, ζs)
= σi
nh
l=1
U
(0)
n−1(ζl, ζs)H + kn
n−
η=n−−µµ
i,η
nh
l=1
f(U−
η )l(ζl, ζs)H,
(2.43)
for s = 1, ..., nh.
We denote the mass matrix M ∈ Rnh × Rnh by
Ml,s := (ζl, ζs)H; (2.44)
also, the stiffness matrix S ∈ Rnh × Rnh is defined by
Sl,s := a(ζl, ζs), (2.45)
which leads to
r
j=0
i,jMUj
n + kn
r
j=0
βi,jSUj
n = σiMU
(0)
n−1 + kn
n−
η=n−−µ
i,ηMf(U−
η ) (2.46)

problems 25
for i = 0, 1, ..., r. This system is used to find the solution on the time interval In
where n ≥ µ + . That is, n ≥ µ in the fully implicit nonlinear case and n ≥ µ + 1
in the implicit–explicit (IMEX) case. The matrix form of the general case of the
system in (2.46) and the matrix forms for the fully-implicit and implicit–explicit
cases can be found in Appendix A.
The integrals in (2.34) are evaluated by using appropriate quadrature rules. Dif-
ferent choices of quadrature formulas can be used depending on the specific appli-
cation. Also, the integrands in (2.35) and (2.36) are polynomials of degree 2r − 1
and 2r, respectively, and can be integrated exactly by using appropriate numerical
quadrature rules.
2.5 hp–dG–timestepping for parabolic systems
We now study the variational discretisation of a semilinear system of evolutionary
parabolic equations in the form: find u, v : I × Ω −→ R such that
∂u
∂t
− l1∆u = f(u, v), in I × Ω,
∂v
∂t
− l2∆v = g(u, v), in I × Ω,
u(t, x) = 0, v(t, x) = 0, for t ∈ I and x ∈ ∂Ω,
u(0, x) = u0, v(0, x) = v0, for x ∈ Ω,
(2.47)
where l1, l2 are the diffusion coefficients, Ω ⊂ Rd
, d = 1, 2, 3 is a polygonal domain
(polyhedral domain in R3
), R is the field of real numbers and I = [0, T] is a finite
time interval with T > 0 being the final time. The unknowns u = u(t, x), v =
v(t, x) represent the solution at the point (position) x at time t ∈ I. f(u, v), g(u, v)
are smooth functions and u0, v0 are the initial conditions at time t = 0. We will
consider here for simplicity in treatment and exposure the homogeneous Dirichlet
boundary conditions (zero boundary conditions) on the boundary of the domain
Ω. The case of non-essential (Neumann) boundary conditions also follows without
any technical challenge, although it is omitted here for brevity. With respect to

problems 26
the notation used in previous sections, this problem corresponds to A = −∆,
V = H1
0 (Ω) and H = L2(Ω).
The process of finding a solution of a system of equations is the same for each
unknown variable in the system. The same techniques used in the derivation of
the dG time-marching schemes for a single semilinear equation can be extended
easily to the system of two (or more) semilinear equations with just some simple
modifications. Indeed, by following the same steps we used in the previous sections
for obtaining (2.20) for (2.10) we arrive at the following linear system of equations
for the system in (2.47):
r
j=0
i,jMUj
n + l1kn
r
j=0
βi,jSUj
n = σiMU
(0)
n−1 + kn
n−
η=n−−µ
i,ηMf(U−
η , V−
η ),
r
j=0
i,jMVj
n + l2kn
r
j=0
βi,jSVj
n = σiMV
(0)
n−1 + kn
n−
η=n−−µ
i,ηMg(U−
η , V−
η ),
(2.48)
for i = 0, 1, ..., r. The matrix form of the general case of the system in (2.48) and
the matrix forms for the fully-implicit and implicit–explicit cases can be found in
Appendix A.
Remark 2.8. The process of finding the approximate solutions to single linear
parabolic equations or systems of linear parabolic equations is similar to the case of
single semilinear parabolic equations or systems of semilinear parabolic equations.
The only difference is that the nonlinear source term f(U) is replaced by the
linear source term f(t, x) and by using appropriate quadrature rules the integrals
involved the source function f can be computed easily.
2.6 Numerical examples and applications
Reaction-diffusion systems are very popular as mathematical models in a wide
range of applications in mathematical biology and mathematical ecology, such
as population dynamics and modelling of biological processes. Typically, these
models are nonlinear and, in particular, semilinear parabolic PDE problems. The

problems 27
solutions to such problems are usually impossible or very difficult to find ana-
lytically. Hence, the alternative is to compute these solutions approximately by
using numerical methods such as the finite element methods. In this section, we
will consider the numerical solution to special cases of these problems modelling
cyclic competition between different species by using the time–discontinuous and
space–continuous Galerkin finite element methods presented above. The numer-
ical implementation is based on the deal.II finite element library [16] and the
tests run in the high performance computing facility ALICE at the University of
Leicester.
2.6.1 Example 1: Fisher system
We consider the solution of the following semilinear reaction-diffusion system
∂u
∂t
− ∆u = f(u, v), in I × Ω,
∂v
∂t
− ∆v = g(u, v), in I × Ω,
u = v = 0, for t ∈ I and x ∈ ∂Ω,
u(0, x) = u0, v(0, x) = v0, x ∈ Ω,
(2.49)
where x = (x, y), for I × Ω = [0, 1] × [0, 1]2
, and the nonlinearities are given by
f(u, v) = u(1 − v) + f1(t, x, y),
g(u, v) = v(1 − u) + g1(t, x, y),
and f1, g1 are independent of the solution components u and v. The initial condi-
tions and boundary conditions are chosen such that the exact solution is:
u = e−t
x(1 − x)y(1 − y),
v = e−2t
x(1 − x)y(1 − y).
We use a rectangular mesh consisting of 1024 uniform biquadratic elements in
space (p = 2) and uniform linear elements in time r = 1, which we denote for

problems 28
brevity as dG(1)–cG(2) scheme. Given that the solution components are quadratic
polynomials in space, this ensures that the space error is negligible and conse-
quently the time error dominates. This allows for assessing the order of conver-
gence of the dG timestepping method by varying the timestep size kn while the
mesh size is kept fixed at h = 1/32. In particular, we study the asymptotic be-
haviour of the error e = u−U in the L∞(L∞)–, L2(L2)–, and L∞(L2)– error norms
by monitoring the evolution of the experimental order of convergence (EOC) over
time on a sequence of uniformly refined meshes in time. We also examine the
superconvergence of the L2–error at the endpoints of the time intervals, denoted
by ∞(L2)–error. The resulting errors are plotted against the corresponding time
step size kn. The EOC of a given sequence of positive quantities ai defined on a
sequence of meshes of step sizes bi is defined by
EOC(a, i) =
log(ai/ai−1)
log(bi/bi−1)
. (2.50)
We report the EOC relative to the last computed quantities in the figure as an
indication of the asymptotic rate of convergence. In this example, ai represent the
error norms and bi are the time step sizes kn.
In Figure 2.1 (a), (b) and (c) we report the L∞(L∞)– L2(L2)– and L∞(L2)–norm
errors, respectively, all of which are of optimal order of convergence with EOC ≈ 2.
Also, Figure 2.1 (d) shows that the superconvergence of the ∞(L2)– error norm at
the endpoints of the time intervals with EOC ≈ 3. The results are in agreement
with theoretical results in Chapter 4.

problems 29
(a) (b)
(c) (d)
Figure 2.1: Example 1: Convergence history for dG(1)–cG(2) scheme for
solving Fisher System.
2.6.2 Example 2: Cycling Lotka–Volterra competition sys-
tem
We solve the semilinear system of three–species competition consisting of three
semilinear parabolic equations with homogeneous Neumann boundary conditions
∂u1
∂t
− D1∆u1 = α1u1(1 − a1,1u1 − a1,2u2 − a1,3u3),
∂u2
∂t
− D2∆u2 = α2u2(1 − a2,1u1 − a2,2u2 − a2,3u3),
∂u3
∂t
− D3∆u3 = α3u3(1 − a3,1u1 − a3,2u2 − a3,3u3),
(2.51)

problems 30
with the initial conditions
u0
1 =
1
1 + e(−γ(x +
√
3 min(y ,0)))
,
u0
2 =
1
1 + e(γ(x −
√
3 min(y ,0)))
,
u0
3 = 1 −
1
1 + e(−γ(y +1/
√
3|x |))
,
(2.52)
where u1, u2, u3 are the densities (concentrations) of the three species at time t
and position (x, y), D1, D2, D3 are the constant diffusion coefficients of these three
species, respectively, and α1, α2, α3 represent the intrinsic growth rates of the three
species, respectively. The coefficients ai,j, i, j = 1, 2, 3, model the limiting effect
that the presence of species uj, j = 1, 2, 3 has on species ui, i = 1, 2, 3. In
particular 1
ai,i
is the carrying capacity of species i, i = 1, 2, 3. The parameter γ
is called the marginal factor and x , y are the shifted coordinates of x, y where
x = x − 0.7L, y = y − 0.7L, where L is the length of the space domain.
We consider the time domain I = [0, 100] and the spatial domain Ω = [0, 150]2
.
The coefficients and parameters have the following values:
D1 = D2 = D3 = 1,
α1 = α2 = α3 = 1,
a1,1 = a1,2 = a2,3 = a3,1 = 1,
a1,3 = a2,1 = a3,2 = 2,
L = 150, γ = 0.5.
We solve the problem by using a dG time stepping method with conforming contin-
uous finite element in space dG(r)-cG(p) with r = 1 and p = 1, 2, on a rectangular
mesh consisting of 4096 uniform biquadratic elements with 4225 and 16641 degrees
of freedom in space, respectively. The time step size is kn = 0.01 resulting in 10000
time steps and 49923 degrees of freedom in time. The numerical solution is shown
in Figure 2.2. For more details see [86, 1, 31]. The solution’s fine scales need high
order numerical schemes for solving such problems with high accuracy.

problems 31
(a) (b)
Figure 2.2: Example 2: The solution at the final time T = 100: u1 in yellow,
u2 in blue, and u3 in red: (a) dG(1)–cG(1), (b) dG(1)–cG(2).
2.6.3 Example 3: Predator–prey system
In this example, we consider the solution for the predator–prey system consist-
ing of two semilinear parabolic equations with homogeneous Neumann boundary
conditions
∂u
∂t
− ∆u = γu(u − β)(1 − u) −
uv
1 + αu
,
∂v
∂t
− ∆v =
uv
1 + αu
− δv,
(2.53)
with the initial conditions
u0 =



p, if |x − L/2| ≤ 11 and |y − L/2| ≤ 12,
0, otherwise,
v0 =



q, if |x − L/2 − a| ≤ 21 and |y − L/2 − b| ≤ 22,
0, otherwise.
Here, u and v are the dimensionless densities (concentrations) of the prey and
predator at time t and position (x, y), = D2
D1
is the ratio of the diffusion coef-
ficients, where D1 is the diffusion coefficient of the prey and D2 is the diffusion

problems 32
coefficient of the predator, α, β, γ, δ are ecological parameters and L is the length
of the space domain.
We consider the time domain I = [0, 163.46] and the spatial domain Ω = [0, 200]2
.
The coefficients and parameters have the following values:
= 1, α = 0.2, β = 0.1, γ = 3, δ = 0.37, p = 1, q = 0.5,
L = 200, a = 5, b = 30, 11= 12= 21= 22= 20.
The mesh in space is rectangular and consists of 4096 uniform biquadratic elements
with 16641 degrees of freedom, and for the time discretisation, we use first degree
polynomials (linear elements) with time step size kn = 0.01 and 16346 time steps
resulting in 49923 degrees of freedom in time. Figure 2.3 shows the generation
(a) (b) (c)
Figure 2.3: Example 3: The solution at the final time T = 163.46: (a) The
Prey, (b) The Predator, (c) The Prey and the Predator superimposed on the
same plot.
of periodical concentric rings for the interactions of the spiral ends. The existing
results of this example are typically obtained by using low order time stepping
schemes and, in particular explicit time stepping schemes. The advantages of
using dG timestepping in solving these problems is the combination of high order
of accuracy and the possibility of using large time steps due to the method being
implicit in the elliptic operator so that we can solve the problem without any
restrictions on the time step size. We refer to [89] for more details on this particular
model.

Chapter 3
A posteriori error analysis
3.1 Introduction
A posteriori error analysis plays an important role in developing and devising
efficient adaptive algorithms which can lead to significant reduction in the compu-
tational costs of approximating the solutions by the numerical methods and this is
a crucial property of any reliable numerical scheme. In a posteriori error analysis
we are interested in finding bounds in the form
||e|| = ||u − U|| ≤ E[U, f, h, kn],
for some function E depending on the approximate solution U and the right hand
side f of the underlying problem, in the relevant norm . , but it does not depend
on the exact solution u of the the underlying problem. The estimator E is an
approximation of the error in the relevant norm if ||e|| ≈ E. Also, it depends on
the data of the problem, mesh step size h and the time step size kn (discretisation
parameters).
The main used techniques for obtaining a posteriori and a priori error bounds
are the energy, duality, and reconstruction. In the energy technique, the error
representation formula is tested with the error or any quantity of interest related
to the error such as temporal or spatial derivatives or integrals of it. The duality
33

Chapter 3 A posteriori error analysis 34
approach depends upon estimating the stability factor analytically or computa-
tionally by solving and using the stability properties of the linear backward dual
problem. For linear PDEs this approach is sharp and for nonlinear PDEs as in
our case (semilinear problems) the analysis relies on stability properties of the
linearised dual problem, and in this case, special care is needed to deal with the
strong stability of the linearised problem. Optimal orders can be obtained in the
L∞(0, T; L2(Ω)) norm by using the duality approach but some tight restrictions
on the spatial mesh have to be imposed. The other disadvantage to this approach
is that no error estimates can be obtained for the gradients. This technique was
introduced by Johnson in 1991, see [43]. For more details about this approach see
[43, 43, 40, 46].
It is a well known fact that the energy technique for parabolic problems results
in optimal rates in L2(0, T; H1
(Ω)) norm and suboptimal rates in L∞(0, T; L2(Ω))
and L2(Ω) norms, but by combining it with construction technique we can retrieve
the optimality in these error norms. The other advantage of the energy technique
is that it enables us to treat nonlinearities with ease. In our analysis we will use
the energy and reconstruction approaches with a continuation argument for energy
estimates in deriving our a posteriori error bounds. For more details about these
issues, see [83, 78].
The reconstruction technique allows us to derive optimal error bounds for higher
order methods for both linear and nonlinear problems with reasonable and prac-
tical assumptions. Also, this technique is ﬂexible and can be used with both the
energy and duality approaches. In the reconstruction technique the estimator E
has four appealing features: (i) E is a computable quantity and depends only on
the approximate solution U and the data of the problem; (ii) If E is not com-
putable then it can be bounded by a bounded quantity; (iii) E is of optimal order
and requires lowest possible regularity; (IV) E contains computable and explicit
stability constants specially for linear problems. For more details see [81].
An error estimator is reliable in the relevant norm if there exists κ1 > 0, indepen-
dent of the exact solution u, satisfying ||e|| ≤ κ1E. E is eﬃcient if there exists
κ2 > 0, independent of the exact solution u, such that κ2E ≤ ||e||. Since these

constants can not be computed explicitly, this motivates the notion of an effectiv-
ity index EI and inverse effectivity index IEI. The effectivity index is defined as
the ratio of the estimator E to ||e|| i.e.
EI :=
E
||e||
,
and
IEI :=
||e||
E
.
Furthermore, the estimator E is robust if the constants κ1 and κ2 do not depend
on the discrete finite element solution U, data of the problem and discretisation
parameters, and it is asymptotically robust if it is robust when the discretisation
parameters are sufficiently small [3, 112].
We will derive a posteriori error estimates in L∞(L2) and L2(H1
) norms for fully
discrete IMEX space–time finite element methods. We use an implicit–explicit
(hp–version) dG timestepping scheme with conforming finite elements in space.
We will derive these a posteriori bounds for the semilinear initial value problem
defined in (2.10) in Chapter 2.
We recall from Section 2.3.3 in Chapter 2 that the fully–discrete IMEX space–time
scheme reads: find U ∈ X such that
In
((U , v)H + a(U, v)) dt + ([U]n−1, v+
n−1)H =
In
(Πf(U), v)H dt (3.1)
for all v ∈ Xn, for n = µ + , ..., N. Depending on the choice of the interpolant
Πf(U) defined in (2.21) we have two cases: the fully implicit scheme and the
implicit–explicit (IMEX) scheme. Despite the specific choices discussed earlier,
in what follows, we shall endeavour to be general with respect to the particular
approximation of the nonlinear term. To that end, we shall refrain from using
specific properties of any particular interpolant/extrapolant used in the proof of
the a posteriori error bounds below, in an effort to be versatile in the choice of
linearisation. Indeed, the a posteriori error bounds given below will involve the
computable quantity Πf(U) − f(U).

3.2 Reconstructions
We now discuss the space-time reconstruction technique proposed in [60] for linear
parabolic problems, which is a combination of the concepts of elliptic reconstruc-
tion for the spatial discretisation [83, 77] and of the dG-timestepping reconstruc-
tion presented first in [84], and further analysed in the hp-setting in [98].
3.2.1 Elliptic reconstruction
For each conforming finite element space Vn ⊂ V, we define the respective dis-
crete elliptic operator An : Vn → Vn to be the unique linear operator such that
(Anw, v)H = a(w, v), for all v, w ∈ Vn.
Given U(t) ∈ Xn, n = 0, ..., N, for t ∈ In, the elliptic reconstruction ˜U(t) =
˜RU(t) ∈ Yn of U is defined as
a( ˜U(t), v) = (AnU(t), v)H, for all v ∈ V, and t ∈ In. (3.2)
The relation (3.2) can be written in pointwise form as A ˜U(·, t) = AnU(·, t), for
all t ∈ In. The reconstruction operator ˜R : X → Y can be represented as ˜R|In =
A−1
An : Vn → V for all n = 0, 1, . . . , N; we refer to [83, 77, 60] for details.
From the definition of An and from (3.2), we have
a( ˜U(t), w) = (AnU(t), w)H = a(U(t), w), for all w ∈ Vn, (3.3)
and, hence, we have
U = ˜Pn
˜U, (3.4)
at each t ∈ In. That is, U is the elliptic projection of the elliptic reconstruction ˜U.
In other words, U is the approximate solution of the elliptic problem whose exact
solution is the elliptic reconstruction function ˜U. Therefore, a crucial consequence
of this construction is the ability to estimate the difference ˜U − U by a posteriori
error estimators for elliptic problems in various norms available in the literature.

As we prefer to keep the exposition independent of specific choices of a posteriori
error bounds for elliptic problems, we opt for merely postulating their existence.
To account for mesh–change effects, we also define the smallest common superspace
V⊕
n := Vn−1 + Vn, and the largest common subspace Vn := Vn−1 ∩ Vn, for all
n = 1, . . . , N.
We introduce the H–projection operator P : V∗
→ V defined by
(Pv, χ)H = v, χ for all χ ∈ V; (3.5)
if we replace V by one of Vn, V⊕
n or Vn in the above definition, the corresponding
H–projection operators are denoted by Pn, P⊕
n or Pn , respectively. Also, we define
the elliptic projection operator ˜Pn : V → Vn by
a( ˜Pnv, w) = a(v, w) for all w ∈ Vn, (3.6)
with ˜Pn the respective elliptic projection onto Vn .
For w ∈ H, we define the time lifting operator Ln : H → Prn
(In; H), by
In
(Ln(w), v)H dt = (w, v+
n−1)H for all v ∈ Prn
(In; H). (3.7)
If W ⊂ H is a linear subspace of H, we have the property
w ∈ W implies Ln(w) ∈ Prn
(In; W); (3.8)
for more details, we refer to [98].
Assumption 3.1 (Elliptic a posteriori error bounds). Let w ∈ V be the exact
solution of the elliptic problem Aw = g with respective boundary conditions and
let W ∈ Vh ⊂ V be the finite element solution of this problem in the finite element
space Vh. We assume that there exist a posteriori error bounds
w − W S ≤ ES[W, g], (3.9)

for S ∈ {H, V, V∗
}.
The literature for such elliptic a posteriori error bounds is vast; see, e.g., [3, 9,
112, 11, 10] and the references therein.
Proposition 3.2 (Dual norm estimate). Let Vh ⊂ V be a (conforming) finite
element space and let Ah the respective discrete elliptic operator defined by Ah :
Vh → Vh such that (AhW, V )H = a(W, V ), for all V, W ∈ Vh. For any v ∈ V∗
,
defining the function ξ as ξ := A−1
h Pv, we have the bound
v 2
V∗ ≤ ˜α2
EV[ξ, v] + ˜α(Pv, ξ)H. (3.10)
where ˜α > 0 is such that v V∗ ≤ ˜α v H.
Proof. For the proof, we refer to [60].
In particular, Assumption 3.1 will imply the validity of the estimates
˜U − U S ≤ ES[U, AnU], S ∈ {H, V, V∗
}, (3.11)
among other things; which are presented in Proposition 3.8 below for details.
By replacing Vn with V⊕
n or by Vn , we signify the corresponding discrete operators
A⊕
n or An , and we denote by ˜R⊕
n or by ˜Rn the respective elliptic reconstructions.
Using (3.3), the IMEX method (3.1) can be re-written as
In
(U , v)H + a( ˜U, v) dt + ([U]n−1, v+
n−1)H =
In
(Πf(U), v)H dt (3.12)
for all v ∈ Xn, for n = 1, . . . , N.
3.2.2 Time reconstruction of ˜U
We define the time reconstruction function Û ∈ H1
(0, T; H) of the elliptic re-
construction ˜U ∈ Y (of the approximate solution U,) as follows: for each In,

n = 1, . . . , N,
Û|In ∈ Prn+1
(In; V), n = 1, . . . , N, (3.13)
satisfies
In
( Û , v)H dt =
In
( ˜U , v)H dt + ([ ˜U]n−1, v+
n−1)H for all v ∈ Yn, (3.14)
and
Û+
n−1 =



u0, n = 0;
˜U−
n−1, n = 1, . . . , N.
(3.15)
The time reconstruction Û is well-defined: we have rn + 2 unknowns per time
interval In and rn + 1 conditions from (3.14) and one more condition from (3.15).
The time reconstruction is also unique and globally continuous with respect to the
time variable as shown in the following lemma. This property is useful in deriving
a pointwise perturbed differential equation for the error (the error representation
formula), or a part thereof. Also, it allows us to use the continuation argument
in the a posteriori error analysis. We note that, the time reconstruction Û is a
higher order reconstruction (polynomial in time on the time interval In), and it is
one degree higher than the elliptic reconstruction ˜U. We finally note from (3.14)
and (3.15) that the time reconstruction is constructed (elementwise) locally.
Equivalently, using the lifting operator (3.7), we can define Û|In ∈ Prn+1
(In; V)
on each time interval In, n = 1, . . . , N, by
Û|In (t) :=
t
tn−1
˜U + Ln([ ˜U]n−1) dτ + ˜U−
n−1, (3.16)
where we recall that ˜U−
0 := u0. For convenience, we also encode the time recon-
struction process as an operator ˆR|In : Prn
(In; V) → Prn+1
(In; V), n = 1, . . . , N.
Hence, we have Û = ˆR ˜U.
Lemma 3.3 (Continuity of the time reconstruction). The time reconstruction,
which is uniquely defined in (3.14) and (3.15), is globally continuous.

Proof. By integrating by parts the left–hand side of (3.14), we see
In
( Û , v)H dt = −
In
( Û, v )H dt + ( Û−
n , v−
n )H − ( Û+
n−1, v+
n−1)H, ∀v ∈ Yn. (3.17)
Now, integrating by parts the right–hand side of (3.14), we find
In
( ˜U , v)H dt + ([ ˜U]n−1, v+
n−1)H = −
In
( ˜U, v )H dt + ( ˜U−
n , v−
n )H
− ( ˜U−
n−1, v+
n−1)H, ∀v ∈ Yn. (3.18)
From (3.17) and (3.18) we have
−
In
( Û, v )H dt + ( Û−
n , v−
n )H − ( Û+
n−1, v+
n−1)H =
−
In
( ˜U, v )H dt + ( ˜U−
n , v−
n )H − ( ˜U−
n−1, v+
n−1)H, ∀v ∈ Yn. (3.19)
Hence,
−
In
( Û, v )H dt + ( Û−
n , v−
n )H = −
In
( ˜U, v )H dt + ( ˜U−
n , v−
n )H, ∀v ∈ Yn, (3.20)
since Û+
n−1 = ˜U−
n−1. By choosing v constant in time we obtain
( Û−
n , v)H = ( ˜U−
n , v)H, ∀v ∈ Yn. (3.21)
Consequently we get
Û−
n = ˜U−
n . (3.22)
Hence, Û is a globally continuous function.
Proposition 3.4 (Time reconstruction error bounds). Let S ⊆ H and Ψ ∈
Prn
(In; S), for n = 1, ..., N. Then, we have the identities:
ˆΨ − Ψ L2(In;S) = Cn [Ψ]n−1 S, S ∈ {H, V, V∗
}, (3.23)
with
Cn :=
kn(rn + 1)
(2rn + 1)(2rn + 3)
1/2
,

and
ˆΨ − Ψ L∞(In;S) = [Ψ]n−1 S, (3.24)
where ˆΨ is defined by
In
(ˆΨ , v)H dt =
In
(Ψ , v)H dt + ([Ψ]n−1, v+
n−1)H for all v ∈ Yn,
and ˆΨ+
n−1 = Ψ−
n−1, n = 1, ..., N and Ψ−
0 given.
Proof. The proof of (3.23) first appeared in [84, Lemma 2.2]; the formula for Cn
was further refined to be explicit in the dependence on rn in [98, Theorem 2].
3.3 A posteriori error bounds
We begin by decomposing the error as
e := u − U = (u − Û) + ( Û − ˜U) + ( ˜U − U) = ρ + σ + .
Note that σ is the time reconstruction error which can be estimated using Propo-
sition 3.4. Similarly, is the elliptic reconstruction error and, therefore, can be
estimated using Assumption 3.1. Thus, it remains to estimate ρ by quantities in-
volving the problem data and/or σ and . To do so, we shall work with energy esti-
mates, in conjunction with a continuation argument to treat the non-Lipschitzian
nonlinear reactions.
From (3.12) and the definition of the time reconstruction (3.14), (3.15), we deduce
In
( Û , v)H + a( ˜U, v) dt
=
In
(Πf(U), v)H + ( , v)H dt + ([ ]n−1, v+
n−1)H, for all v ∈ Xn,
(3.25)
which can be written in pointwise form as
Pn
Û + A Û = PnΠf(U) + Pn + Ln([ ]n−1) + Aσ, (3.26)

n = 1, . . . , N. Subtracting (3.26) from (2.10), we obtain
ρ + Aρ = f(u) − PnΠf(U) + Pn
Û − Û − Pn + Ln([ ]n−1) − Aσ, (3.27)
for n = 1, . . . , N. From (3.16), we deduce that Û = ˜U +Ln([ ˜U]n−1) and, therefore,
we can arrive at
Pn
Û − Û − Pn + Ln([ ]n−1) = − − Ln([ ]n−1) + Ln(U−
n−1 − PnU−
n−1),
upon observing that PnU = U in In. Using this identity in (3.27) we arrive at
an error equation for ρ:
ρ + Aρ = f(u) − PnΠf(U) − − Ln([ ]n−1) + Ln(U−
n−1 − PnU−
n−1) − Aσ,(3.28)
on which we can now apply energy–type arguments.
For brevity, we set P : [0, T] → V, defined as P|In = Pn, n = 0, . . . , N; we shall
use the corresponding notation L(v) to denote collectively the liftings on each
time interval, and so, L(v)|In = Ln(vn−1), n = 1, . . . , N. Also, we denote by
emc : [0, T] → V the error due to the mesh change between the finite element
spaces Vn−1 and Vn given by emc|In := Ln(U−
n−1 − PnU−
n−1), n = 1, . . . , N.
We test (3.28) with ρ, integrate in space and in time between 0 to t ∈ I, we deduce
1
2
ρ(t) 2
H +
t
0
a(ρ, ρ) dτ =
t
0
(f(u) − PΠf(U), ρ)H dτ +
t
0
(emc, ρ)H dτ
−
t
0
( + L([ ]), ρ)H dτ −
t
0
a(σ, ρ) dτ,
(3.29)
noticing that ρ(0) = 0 by construction. Employing the coercivity (2.13) and
continuity (2.12) of a, the last estimate implies
1
2
ρ(t) 2
H + Ccoer
t
0
ρ 2
V dτ ≤
t
0
(f(u) − PΠf(U), ρ)H dτ
+
t
0
(emc, ρ)H dτ −
t
0
(D , ρ)H dτ − Ccont
t
0
σ V ρ V dτ, (3.30)

upon introducing the notation D := + L([ ]). Using Young inequality for the
third and fourth terms on the right hand side of (3.30), implies that
1
2
ρ(t) 2
H + 1 − γ Ccoer
t
0
ρ 2
V dτ ≤
t
0
+
t
0
(emc, ρ)H dτ +
1
2γCcoer
t
0
D 2
V∗ + C2
cont σ 2
V dτ, (3.31)
for any γ > 0. The second term on the right-hand side of (3.30) can be further
estimated by
t
0
(emc, ρ)H dτ ≤ λ ρ L∞(0,t;H)
t
0
emc H dτ + (1 − λ)
t
0
emc V∗ ρ V dτ
≤
sign λ
4
ρ 2
L∞(0,t;H) + λ2
t
0
emc H dt
2
+
(1 − λ)2
Ccoer
t
0
emc
2
V∗ dτ + sign(1 − λ)
Ccoer
4
t
0
ρ 2
V dτ,
(3.32)
for any 0 ≤ λ ≤ 1, with the sign denoting a sign function where, in particular,
sign ν = 0 if ν = 0. An interesting choice is λ := min{1, t−1/2
}, in that it can
counteract the imbalance caused by the L1
-accumulation of the error on the second
term on the right-hand side of (3.32):
λ2
t
0
emc H dτ
2
≤ λ2
t
t
0
emc
2
H dτ ≤
t
0
emc
2
H dτ,
thereby retaining a dimensional balance in the context of long–time simulations;
we refer to [60, Remark 4.10] for a related discussion. The accumulation of the
mesh change error can be of importance in practical simulations [39], as it accounts
for the loss of information caused by the mesh modiﬁcation.
Selecting now γ = γλ := 1/2 − sign(1 − λ)/4 in (3.30) and using (3.32), we arrive
at
ρ(t) 2
H + Ccoer
t
0
ρ 2
V dτ ≤ 2
t
0
+ 2λ2
emc
2
L1(0,t;H) +
2(1 − λ)2
Ccoer
emc
2
L2(0,t;V∗)
+
1
γλCcoer
D 2
L2(0,t;V∗) + C2
cont σ 2
L2(0,t;V)
+
sign λ
2
ρ 2
L∞(0,t;H).
(3.33)

To estimate the last term on the right-hand side of (3.33), we return to (3.30)
setting γ = 2γλ to deduce
1
2
ρ(t) 2
H ≤
t
0
+
1
4γλCcoer
D 2
L2(0,t;V∗) + C2
cont σ 2
L2(0,t;V)
+ λ2
emc
2
L1(0,t;H) +
(1 − λ)2
Ccoer
emc
2
L2(0,t;V∗)
+
sign λ
4
ρ 2
L∞(0,t;H) := (I) +
sign λ
4
ρ 2
L∞(0,t;H).
(3.34)
Now setting t = t∗
such that ρ(t∗
) H = ρ L∞(0,t;H) in (3.34), we deduce
2 − sign λ
4
ρ 2
L∞(0,t;H) ≤ (I), or
sign λ
2
ρ 2
L∞(0,t;H) ≤
2 sign λ
2 − sign λ
(I), (3.35)
which we use to bound the last term on the right-hand side of (3.34) further and,
by adding ρ 2
L∞(0,t;H) ≤ 4/(2 − sign λ)(I) to the resulting estimate, we arrive
ﬁnally at
ρ 2
L∞(0,t;H) + Ccoer ρ 2
L2(0,t;V) ≤
c2,λ
Ccoer
D 2
L2(0,t;V∗) + C2
cont σ 2
L2(0,t;V)
+ c1,λ
t
0
|(f(u) − PΠf(U), ρ)H| dτ
+ λ2
emc
2
L1(0,t;H) +
(1 − λ)2
Ccoer
emc
2
L2(0,t;V∗)
(3.36)
with c1,λ = 4 for λ = 0 and c1,λ = 8 for 0 < λ ≤ 1, and c2,λ = 4 if λ = 0,
c2,λ = 16/3 if 0 < λ < 1, and c2,λ = 8/3 if λ = 1.
We shall now estimate each term on the right–hand side of (3.36) separately.
3.3.1 Estimating the nonlinear term
We decompose the integrand in the nonlinear term in (3.36) as
(f(u) − PΠf(U), ρ)H ≤ (f(u) − f(U), ρ)H + f(U) − PΠf(U) V∗ ρ V, (3.37)
with f(U) − PΠf(U) V∗ measuring how well PΠf(U) approximates f(U).

As we shall make use of the Sobolev Imbedding Theorem, the discussion in this
section comes under the speciﬁc choice H = L2(Ω) and V = H1
0 (Ω); the case of
non-essential boundary conditions also follows without any technical challenge,
although it is omitted here for brevity.
Lemma 3.5 (Estimation of the nonlinear term). If the nonlinear reaction f is as
in Section 3.1, satisfying the growth condition (2.14) with 0 ≤ r < 2 for d = 2,
and with 0 ≤ r ≤ 4/3 for d = 3, we have the bound
Ω
|f(u) − f(U)||ρ| dx ≤ C ρ r
L2(Ω) ρ 2
L2(Ω) + CG(U) ρ 2
L2(Ω)
+ C σ r
L2(Ω) σ 2
L2(Ω) + r
L2(Ω)
2
L2(Ω)
+ CG(U) σ 2
L2(Ω) + 2
L2(Ω) ,
(3.38)
where G(U) := 1 + U r
L∞(Ω).
Proof. Using the growth condition (2.14), along with the elementary inequality
|a + b|r
≤ C(|a|r
+ |b|r
), we have, respectively,
Ω
|f(u) − f(U)||ρ| dx ≤ C
Ω
|u − U|(1 + |u|r
+ |U|r
)||ρ| dx
≤ C
Ω
|u − U|(1 + |u − U|r
+ |U|r
)||ρ| dx
≤ C
Ω
|u − U|r+1
|ρ| dx
+ C
Ω
(1 + |U|r
)|u − U||ρ| dx.
(3.39)
For the ﬁrst term on the right–hand side of (3.39) we use the inequality
Ω
|v|r+1
|w| dx =
r + 1
r + 2
v r+2
Lr+2(Ω) +
1
r + 2
w r+2
Lr+2(Ω), (3.40)
thereby, deducing
Ω
|u − U|r+1
|ρ| dx ≤ C( ρ r+2
Lr+2(Ω) + σ r+2
Lr+2(Ω) + r+2
Lr+2(Ω)). (3.41)

Recalling the assumption 0 ≤ r < 2, Hölder’s inequality with exponent p = 2/r,
(and, thus, q = 2/(2 − r),) we have
ρ r+2
Lr+2(Ω) =
Ω
|ρ|r
|ρ|2
dx ≤ ρ r
L2(Ω) ρ 2
L4/(2−r)(Ω) ≤ C ρ r
L2(Ω) ρ 2
L2(Ω), (3.42)
using the Sobolev Imbedding Theorem ρ L4/(2−r)(Ω) ≤ CS ρ L2(Ω), with 0 ≤ r <
2 for d = 2 and 0 ≤ r ≤ 4/3 for d = 3. Similarly, we have the same estimate
(3.42), with ρ replaced by σ and .
Now, the second term of (3.39) can be dealt with as follows
Ω
(1 + |U|r
)|u − U||ρ| dx ≤
Ω
(1 + |U|r
) |ρ|2
+ |σ||ρ| + | ||ρ| dx
≤
Ω
(1 + |U|r
) 2|ρ|2
+
1
2
|σ|2
+
1
2
| |2
dx
≤ (1 + U r
L∞(Ω)) 2 ρ 2
L2(Ω) +
1
2
σ 2
L2(Ω) +
1
2
2
L2(Ω) .
(3.43)
Combining the above estimates, we arrive at the required bound.
To retain the abstract and more compact notation from the previous section, we
write (3.38) as follows
(f(u) − f(U), ρ)H ≤ C ρ r
H ρ 2
V + G(U) ρ 2
H
+ σ r
H σ 2
V + r
H
2
V
+ G(U) σ 2
H + 2
H ,
(3.44)
and we assume its validity henceforth for any H and V.
3.3.2 Completing the estimate
The bound of the nonlinear term (3.44) still contains norms of ρ on the right-hand
side. To eliminate these, we shall employ a continuation argument in the spirit of
[19, 28, 30].

To this end, using Lemma (3.5) to bound the respective term on the right–hand
side of (3.36), we arrive at
ρ 2
L∞(0,t;H) +
Ccoer
2
t
0
ρ 2
V dτ ≤ E1(t, U, σ, ) + C
t
0
ρ r
H ρ 2
V dτ
+ C
t
0
G(U) ρ 2
H dτ,
(3.45)
where
E1(t, U, σ, ) :=
c2,λ
Ccoer
D 2
L2(0,t;V∗) + C2
cont σ 2
L2(0,t;V)
+
2c1,λ
Ccoer
f(U) − PΠf(U) 2
L2(0,t;V∗)
+ c1,λ λ2
emc
2
L1(0,t;H) +
(1 − λ)2
Ccoer
emc
2
L2(0,t;V∗)
+ C
t
0
σ r
H σ 2
V + r
H
2
V + G(U) σ 2
H + 2
H dτ.
(3.46)
Upon observing that
t
0
ρ r
H ρ 2
V dτ ≤ ρ r
L∞(0,t;H)
t
0
ρ 2
V dτ
≤ ρ 2
L∞(0,t;H) +
t
0
ρ 2
V dτ
1+r
2
≤ C ρ 2
L∞(0,t;H) +
Ccoer
2
t
0
ρ 2
V dτ
1+r
2
,
(3.47)
we deduce
ρ 2
L∞(0,t;H) +
Ccoer
2
t
0
ρ 2
V dτ ≤ E1(t, U, σ, ) + C1
t
0
G(U) ρ 2
H dτ
+ C2 ρ 2
L∞(0,t;H) +
Ccoer
2
t
0
ρ 2
V dτ
1+ r
2
,
(3.48)
for known constants C1, C2 > 0. For each n = 1, . . . , N, we let δn := E1(tn, U, σ, )
. and consider the interval
Jn := t ∈ [0, tn] : ρ 2
L∞(0,t;H) +
Ccoer
2
t
0
ρ 2
V dt ≤ 4δnF(tn, U) ,
where we set F(tn, U) := exp C1
tn
0 G(U) dτ , for brevity. We observe that Jn = ∅
as ρ 2
L∞(0,t;H) + Ccoer
2
t
0 ρ 2
V dτ is continuous with respect to t and that it is equal
to zero for t = 0, owing to the property ρ(0) = 0; also, Jn is closed.

Assuming, without loss of generality, that r > 0, (for, otherwise, f in (2.10) is
globally Lipschitz continuous and, thus, the a posteriori bounds follow by com-
bining the results from [60] along with a standard Grönwall inequality,) we set
t := max Jn > 0.
Suppose that tn > t , i.e., tn /∈ Jn. Hence, δn = E1(tn, U, σ, ) ≥ E1(t , U, σ, ).
Therefore, (3.48) with t = t yields
ρ 2
L∞(0,t ;H) +
Ccoer
2
t
0
ρ 2
V dτ ≤ δn + C2 4δnF(tn, U)
1+r
2
+ C1
t
0
G(U) ρ 2
H dτ,
(3.49)
and Grönwall inequality, thus, implies
ρ 2
L∞(0,t ;H) +
Ccoer
2
t
0
ρ 2
V dτ ≤ F(tn, U) C2 4δnF(tn, U)
1+ r
2
+ δn ,
(3.50)
since F(tn, U) ≥ F(t , U). Upon assuming that δn is such that
C2 4δnF(tn, U)
1+ r
2
≤ δn, or δn ≤ C
−2/r
2 4F(tn, U)
−2+r
r
,
the estimate (3.50) becomes
ρ 2
L∞(0,t ;H) +
Ccoer
2
t
0
ρ 2
V dτ ≤ 2δnF(tn, U); (3.51)
this is a contradiction, as t was assumed to be the maximum element of Jn. Hence,
tn = t and, thus, we have already proven the following result.
Lemma 3.6. Assuming the validity of estimate (3.44), (or, in the special case of
H = L2(Ω) and V = H1
0 (Ω), assuming the hypotheses of Lemma 3.5,) the following
conditional estimate holds: provided that
E1(tn, U, σ, ) ≤ C
−2/r
2 4F(tn, U)
−2+r
r
, (3.52)
we have the bound
ρ 2
L∞(0,tn;H) +
Ccoer
2
ρ 2
L2(0,tn;V) ≤ 4F(tn, U)E1(tn, U, σ, ). (3.53)

We observe that the condition (3.52) in the estimate above is computable, provided
that E1(tn, U, σ, ) is computable. With this in mind, we shall bound the norms
of σ and in E1 by computable quantities below. Crucially, if δn is computable,
then (3.53) becomes an a posteriori bound for ρ. The triangle inequality, would
then already yield an a posteriori bound for the error e. Of course, we expect
that δn decreases arbitrarily as the maximum timestep and spatial meshsize decay
and/or the order of the dG-timestepping increases. We note, finally, that such
conditional estimates are the “a posteriori equivalents” to the standard smallness
assumptions on timestep and meshsize appearing in a priori error bounds for finite
element methods for nonlinear evolution problems.
Remark 3.7. Crucially, there is no explicit CFL-type restriction in the statement
of Lemma 3.6, despite this being concerned with an IMEX discretisation. Indeed,
for unstable combinations of timesteps and spatial meshsizes, the bound (3.53)
remains valid, provided the condition (3.52) is satisfied. It is, therefore, conceivable
that (3.52) holds for CFL-unstable scenarios also; in such cases, (3.53) will remain
valid, resulting to arbitrarily large right–hand sides, c.f., also [58].
3.3.3 Estimating the norms of σ and of
Proposition 3.8 (Bounds on norms of ). Given Assumption 3.1, if ˜U = RU,
then for t ∈ In, n = 0, 1 . . . , N, we have, for = ˜U − U and D = + L([ ]),
respectively, the bound
S ≤ ηS,n := ES[U, AnU], (3.54)
and
D V∗ ≤ ζV∗,n (3.55)
with
ζV∗,n := EV∗ [ ˜Pn (U + Ln([U]n−1)), AnU + AnLn(U+
n−1) − An−1Ln(U−
n−1)].

Proof. Noting that the elliptic reconstruction ˜U is time–independent and therefore
commutes with time diﬀerentiation, (3.54) follows immediately by (3.2) along with
Assumption 3.1.
Now, observing the identity,
a( ˜U + Ln([ ˜U]n−1), v) = (AnU + AnLn(U+
n−1) − An−1Ln(U−
n−1), v)H, (3.56)
which is valid for all v ∈ V, we have the Galerkin orthogonality property
a( ˜U + Ln([ ˜U]n−1), v) = a(U + Ln([U]n−1), v) for all v ∈ Vn . (3.57)
The above means that the elliptic problem (3.56) has the ﬁnite element solution
˜Pn (U + Ln([U]n−1)) on Vn . In view of Assumption 3.1, (3.55) follows.
It is possible to prove an alternative bound to (3.55) by assuming a Poincaré-
Friedrichs/spectral gap type inequality v H ≤ CPF v V and an a posteriori error
bound in the H–norm. Indeed, if we seek z ∈ V, such that a(v, z) = (D , v)H, and
we assume that z is smooth enough, we have
D 2
H = a(D , z) = a(D , z − Z),
for any Z ∈ Vn from the Galerkin orthogonality (3.57). From this point, one
can work in a standard fashion to arrive at a residual–type (or other) a posteriori
error bound EH utilising the approximation properties of Vn and (any) additional
regularity z ∈ V ⊂ V, say, such that z V ≤ C D H, resulting to a bound of
the form
D H ≤ ζH,n
where
ζH,n := EH(U + Ln([U]n−1), AnU + AnLn(U+
n−1) − An−1Ln(U−
n−1)).

Now,
D V∗ = sup
0=w∈V
(D , w)H
w V
≤ sup
0=w∈V
D H w H
w V
≤ CPF D H,
resulting in the alternative estimate D V∗ ≤ CPFζH,n; cf., also [77] for a related
result in the lowest order case using backward Euler timestepping. This estimate
has the advantage of not requiring the elliptic projection onto Vn be evaluated.
In practice, one can take the minimum of the two estimates
D V∗ ≤ min ζV∗,n, CPFζH,n =: ζmin,n, (3.58)
on In, n = 1, . . . , N, provided they are available. For instance, when H = L2(Ω)
and V = H1
0 (Ω), both estimates in (3.58) are valid.
Proposition 3.9 (Bounds on norms of σ). Given Assumption (3.1), for each In,
n = 0, 1 . . . , N, we have, for σ = ˆU − ˜U, the bounds
σ L2(In;S) ≤ Cn (θS,n + [U]n−1 S) ,
where
θS,n := ES[ ˜Pn [U]n−1, AnU+
n−1 − An−1U−
n−1],
for S ∈ {H, V}, and
σ L∞(In;H) ≤ θH,n + [U]n−1 H.
Proof. From Proposition 3.4, we have
σ 2
L2(In;V) = ˆU − ˜U 2
L2(In;V) = C2
n [ ˜U]n−1
2
V. (3.59)
The triangle inequality implies [ ˜U]n−1 V ≤ [ ]n−1 V + [U]n−1 V. To estimate
[ ]n−1 V, we work completely analogously to the proof of Proposition 3.8: we
observe the Galerkin orthogonality
a([ ˜U]n−1, v) = a([U]n−1, v) for all V ∈ Vn ,

which, together with Assumption 3.1 give rise to the estimate
[ ]n−1 V ≤ EV[ ˜Pn [U]n−1, AnU+
n−1 − An−1U−
n−1].
From (3.24) in Proposition 3.4, we also have
σ L∞(In;H) = [ ˜U]n−1 H ≤ [ ]n−1 H + [U]n−1 H,
which, working as above, gives the second estimate.
For an alternative bound, we refer to [60, Lemma 4.4].
Remark 3.10. If no mesh modification takes place, i.e., when Vn−1 = Vn, the above
estimates simplify considerably, since we then have
θS,n = ES[[U]n−1, An[U]n−1].
Using Propositions 3.8 and 3.9 we can bound the term E1(tn, U, σ, ) given in (3.46)
by E1(tn, U) defined as
E1(tn, U) :=
c2,λ
Ccoer
N
n=1
ζ2
min,n + C2
cont Cn(θV,n + [U]n−1 V)
2
+
2c1,λ
Ccoer
f(U) − PΠf(U) 2
L2(0,tn;V∗)
+ c1,λ λ2
emc
2
L1(0,tn;H) +
(1 − λ)2
Ccoer
emc
2
L2(0,tn;V∗)
+ C
N
n=1
θH,n + [U]n−1 H
r
Cn (θV,n + [U]n−1 V)
+ max
t∈In
ηH,n(t)
r
In
η2
V,n(t) dt
+ max
t∈In
G(U(t)) CnθH,n + Cn [U]n−1 H
2
+
In
η2
H,n(t) dt ,
using which, we are now in a position to finalise the a posteriori error analysis.

3.3.4 The ﬁnal a posteriori error bounds
Using the bounds of ρ, σ and , we are now ready to complete the a posteriori
error analysis.
Theorem 3.11 (L∞(I; H)–norm estimate). Assuming the validity of estimate
(3.44), (or, in the special case of H = L2(Ω) and V = H1
0 (Ω), assuming the
hypotheses of Lemma 3.5,) the following conditional estimate holds: provided that
E1(tn, U) ≤ C
−2/r
2 4F(tn, U)
−2+r
r
, (3.60)
for n = 1, ..., N, we have the a posteriori error bound
u − U 2
L∞(0,tn;H) ≤ 4F(tn, U)E1(tn, U)
+ max
i=1,...,n
θH,i + [U]i−1 H
2
+ max
t∈[0,tn]
η2
H,n.
(3.61)
Proof. We begin by using triangle inequality which implies
u − U 2
L∞(0,tn;H) ≤ ρ 2
L∞(0,tn;H) + σ 2
L∞(0,tn;H) + 2
L∞(0,tn;H). (3.62)
Then by observing that the proof and the statement of Lemma 3.6 holds with
E1(tn, U, σ, ) replaced by E1(tn, U), we have
ρ 2
L∞(0,tn;H) ≤ 4F(tn, U)E1(tn, U). (3.63)
Noting that σ represents the time reconstruction error, then from Proposition 3.4
we obtain
σ 2
L∞(0,tn;H) = ˆU − ˜U 2
L∞(0,tn;H) := max
n=1,...,N
[ ˜U]n−1
2
H. (3.64)
Proposition 3.9 implies that
σ 2
L∞(0,tn;H) ≤ max
i=1,...,n
θH,i + [U]i−1 H
2
. (3.65)

Now, it remains to bound = ˜U − U which is the elliptic error and by the aid of
Proposition 3.8 we obtain
2
L∞(0,tn;H) ≤ max
t∈[0,tn]
η2
H,n. (3.66)
Finally, by substituting (3.63), (3.65), and (3.66) in (3.62) we obtain the result.
Similarly, we have an a posteriori bound in the L2(I; V)–norm.
Theorem 3.12 (L2(I; V)–norm estimate). Assuming the validity of estimate (3.44),
(or, in the special case of H = L2(Ω) and V = H1
0 (Ω), assuming the hypotheses of
Lemma 3.5,) the following conditional estimate holds: provided that (3.52) holds
for n = 1, ..., N, we have the a posteriori error bound
u − U 2
L2(0,tn;V) ≤
6
Ccoer
4F(tn, U)E1(tn, U)
+
N
n=1
C2
n θV,n + [U]n−1 V
2
+
In
η2
V,n dt .
(3.67)
Proof. By the use of the triangle inequality we obtain
u − U 2
L2(0,tn;V) ≤ ρ 2
L2(0,tn;V) + σ 2
L2(0,tn;V) + 2
L2(0,tn;V). (3.68)
Noting that the proof and the statement of Lemma 3.6 holds with E1(tn, U, σ, )
replaced by E1(tn, U), and then we have
ρ 2
L2(0,tn;V) ≤
8
Ccoer
F(tn, U)E1(tn, U). (3.69)
Also, observe that σ is the time reconstruction error, hence from Propositions 3.4
and 3.9 we obtain
σ 2
L2(0,tn;V) ≤
N
n=1
C2
n θV,n + [U]n−1 V
2
. (3.70)
Using Proposition 3.8 to bound the elliptic error we have
2
L2(0,tn;V) ≤
N
n=1 In
η2
V,n dt. (3.71)

Now, substituting (3.69), (3.70), (3.71) in (3.68) leading to the required result.
3.4 Numerical experiments
We present a series of numerical experiments aimed at testing the reliability and
efficiency of the a posteriori error bounds derived above. The numerical imple-
mentation is based on the deal.II finite element library [16] and the tests run in
the high performance computing facility ALICE at the University of Leicester.
We study the asymptotic behaviour in the L∞(L2)– and L2(H1
)–norms of the error
and of the respective estimators by monitoring the evolution of the experimental
order of convergence (EOC) defined in (2.50) over time on a sequence of uniformly
refined space meshes indexed by the mesh size h. In each instance, we fix a
constant time step kn as some power of h and we also use fixed polynomial degrees
in both space and time. The resulting errors and estimators are plotted against
the corresponding space mesh size h.
We report the EOC relative to the last computed quantities in all figures as an
indication of the asymptotic rate of convergence. We also report the respective
effectivity indices, i.e., the ratio between estimator and error for each instance.
The estimator is deemed reliable if the effectivity is greater than or equal to one
and it is most efficient when the effectivity is close to one.
In the examples below we consider both linear and semilinear parabolic problems.
In all cases, A = ∆, i.e., the Dirichlet Laplacian, yielding the heat equation
with either linear or nonlinear source terms and H = L2(Ω), V = H1
0 (Ω), giving
H∗
= H−1
(Ω).
3.4.1 Example 1: a linear problem
We test the IMEX fully discrete scheme analysed in this work on (2.10) with
I × Ω := [0, 1] × [0, 1]2
, f is independent of the exact solution u and the initial

and boundary conditions such that the exact solution is given by
u(t, x, y) = sin(πt) sin(πx) sin(πy). (3.72)
The respective a posteriori error bounds when the PDE is linear can be trivially
recovered from Theorems 3.11 and 3.12 by setting r = 0 and removing the con-
ditionality estimate (3.52) as it is void in the linear case; this can be seen by
observing that the second term on the right–hand side of (3.49) disappears when
the forcing f is a function of t and x only. Alternatively, we refer to [60] for a
thorough treatment of the linear case.
We report the results of two tests using different combinations of polynomial orders
r and p in time and space, respectively, denoted as dG(r)–cG(p) scheme.
3.4.1.1 Example 1A: dG(1)–cG(2) scheme
Here, we employ uniform biquadratic elements in space (p = 2) and uniform
linear elements in time (r = 1), i.e., the dG(1)–cG(2) scheme. Figure 3.1 shows
the convergence history with kn = h (left plot) and with kn = h3/2
(right plot)
for both the L∞(L2)– and L2(H1
)–norms. In the case kn = h, we observe that
the L2(H1
) estimator provides the required order of convergence as EOC ≈ 2, in
close agreement with the corresponding error; the effectivity is in between 2.90
and 8.93. Also the L∞(L2) estimator yields the correct rate as EOC ≈ 3, with
effectivity between 47.41 and 63.41.
For the case kn = h3/2
, we again observe the expected order of convergence of the
L2(H1
)–norm error and estimator, while for the L∞(L2)–norm we have an EOC
of 4.64 and 4.72, respectively, corresponding to the convergence rate expected in
time, thus indicating that the time discretisation error dominates in this case.
The effectivity is approximately 5.28 and 7.16 for the L2(H1
)– and L∞(L2)–norm
estimators, respectively. In both cases the results are in agreement with Theorems
3.11 and 3.12.

Figure 3.1: Example 1A. Convergence history for the dG(1)–cG(2) scheme
with kn = h (left) and kn = h3/2 (right).
Figure 3.2: Example 1B. Convergence history for the dG(2)–cG(2) scheme
with kn = h (left) and kn = h4/3 (right).
3.4.1.2 Example 1B: dG(2)–cG(2) scheme
Here, we consider two diﬀerent relations for the timestep and space meshsize. That
is, kn = h and kn = h4/3
, respectively.
The numerical results corresponding to kn = h are shown in the left plot of Fig-
ure 3.2. We observe that our error estimators provide the expected order of con-
vergence in both the L2(H1
)– and L∞(L2)–norms.
The results obtained with the choice kn = h4/3
are reported on the right plot
of Figure 3.2. Again we observe an optimal experimental order of convergence
as EOC ≈ 2 for both the L2(H1
)–norm estimator and error. The respective

experimental order of convergence of the L∞(L2)–norm estimator and error are
EOC ≈ 4, corresponding to the optimal convergence rate with respect to the
timestep size. In both cases, the estimators’ effectivities show little differences
with the corresponding values obtained in Example 1 and are, therefore omitted
for brevity. Also, the results are in agreement with theoretical results in Theorems
3.11 and 3.12.
3.4.2 Example 2: a nonlinear problem
On I × Ω := [0, 1] × [0, 1]2
we consider the semilinear problem (2.10) with f =
−u2
+ ˜f(x, y, t), with ˜f such that the exact solution is given by
u(t, x, y) = sin(πt) sin(πx) sin(πy); (3.73)
note that we have r = 1 and p = 2 in this case. We test the respective a posteriori
error bounds from Theorems 3.11 and 3.12. We test the dG(1)–cG(2) scheme, by
considering the two choices kn = h and kn = h3/2
with corresponding numerical
results in the left and right plots of Figure 3.3, respectively.
The results are in line with those of the linear example. In particular, for kn = h
we again observe good agreement between the estimators and the corresponding
errors, with EOC ≈ 2 and EOC ≈ 3 for the L2(H1
)– and L∞(L2)– quantities,
respectively.
The results corresponding to kn = h3/2
are also confirming the theoretical asymp-
totic rate of convergence. For the L2(H1
)–norm estimator and error we have
EOC ≈ 2 and, similarly to the linear problem considered earlier, for the L∞(L2)–
norm estimator and error we have EOC ≈ 4.5. Note also that the effectivity is, in
all cases, in between 1.07 and 12.18. We notice that the results coincide with the
results of Theorems 3.11 and 3.12.

Figure 3.3: Example 2. Convergence history for the dG(1)–cG(2) scheme with
kn = h (left) and kn = h3/2 (right).

Chapter 4
A priori error analysis
4.1 Introduction
Determining the quality of the approximate solutions is another interesting area
of research in the study of finite element methods. A priori error bounds are very
helpful and useful tools in this regard. They can be used to judge whether the
numerical solution is close to the exact solution of the problem. In the a priori
error analysis we are interested in bounding the actual error as follows
e = u − U ≤ E[u, f, h, kn],
where the function E depends on the exact solution u and the source term f of the
problem, the mesh size h, the time step size kn, and on the data of the problem,
in the relevant norm . . If this function approaches zero when the mesh is fine
i.e. when h is small, and also for small time steps, then this indicates that the
approximate solution is getting closer and closer to the actual solution. The main
idea in the a prior error analysis is to split up the error in the following form
e = u − U = (u − ˜Phu) + ( ˜Phu − U),
where ˜Ph is the elliptic projection operator, also, known as Wheeler or Ritz projec-
tion, which was first proposed in 1973 by Wheeler [116]. The elliptic reconstruction
60

Chapter 4 A priori error analysis 61
used in the previous chapter is considered as the dual a posteriori of the elliptic
projection in the a priori error analysis. In this section, we will consider the a
priori error analysis in the L∞(L2) and L2(H1
) norms for the fully discrete IMEX
space–time ﬁnite element scheme (2.23) applied to the semilinear evolution model
problem deﬁned in (2.10).
For simplicity we assume that the spatial mesh does not change dynamically.
Let also h : Ω → R denote the elementwise constant meshsize function whereby
h|K = hK, for every spatial element K ∈ Th, with Th denoting the spatial mesh
subordinate to Vh. Throughout this work, we shall assume that crj−1 ≤ rj ≤
Crj−1, where c, C > 0, for all j = 2, . . . N uniformly, i.e., that the polynomial
degrees in the temporal variable admit a local quasi-uniformity condition.
We begin with the following auxiliary result.
Lemma 4.1. For v ∈ C(In; H), and for H = L2(Ω), we have the inverse estimate
v 2
L∞(In;H) ≤ In v 2
gut dt + 2 (4.1)
Proof. letting v ∈ C(In; H) and t∗
∈ In, so that v(t∗
) H = maxt∈In v d, we
have
max
t∈In
v H = v(t∗
) H = −
tn
t∗
d
dt
v(t) H dt + v(t−
n ) /h. (4.2)
Now,
tn
t∗
d
dt
v(t) H dt =
tn
t∗
d
dt Ω
v2
(t, x) dx
1/2
dt
=
1
2
tn
t∗
d
dt Ω
v2
(t, x) dx
Ω
v2
(t, x) dx
−1/2
dt
=
tn
t∗ Ω
v(t, x)v (t, x) dx
Ω
v2
(t, x) dx
−1/2
dt
≤
tn
t∗
v(t) H v (t) H v(t) −1
H dt =
In
v (t) H dt.
Using this in (4.2), upon squaring and using the Cauchy-Schwarz inequality, gives
max
t∈In
v 2
H ≤ 2kn
In
v (t) 2
H dt + 2 v(t−
n ) 2
H.

When H is not the canonical case H = L2(Ω), we make the following assumption
instead.
Assumption 4.2. For v ∈ C(In; H), and for some C > 0, independent of kn and of
v, we have the estimate
v 2
L∞(In;H) ≤ C kn
In
v (t) 2
H dt + v(t−
n ) 2
H . (4.3)
We introduce the space-time projection operator P : L2(I; V) → X by
P := πn
⊗ ˜Ph,
i.e., it is a time-interval-wise L2-orthogonal (discontinuous) projection (πn
) with
respect to the time variable tensorised with the elliptic projection ( ˜Ph) in space,
for some n ∈ {1, . . . , N}. Also, we shall make the (mildly) simplifying assumption
(w, v)V = (Aw, v)H = (
√
Aw,
√
Av)H; (4.4)
we stress, however, that certain generalisations are possible, although not carried
through here for simplicity of the presentation.
The a priori error bounds given below will involve the assumption that the quan-
tity Πf(u) − f(u) is optimally convergent and that Π is stable in suitable norms.
4.2 A priori error bounds
We begin by proving an a priori error bound for the L∞(I; H)– and L2(I; V)–
norms of the error. The proof is based on the combination of hp-version approxi-
mation estimates with an inf-sup condition argument, a variant of which has been
presented already in [26], see also [82], along with known arguments for linear part
of the operator (see, e.g., [109, Chapter 12]). The results presented below extend
the theory from [52] to the case of non-Lipschitz nonlinear reactions.

4.2.1 The stability of Pu − U
For brevity we set ϑ := Pu − U, and we decompose the error as
u − U = (u − Pu) + (Pu − U) =: p + ϑ,
with p := u−Pu. Note that p is a projection error and, therefore, can be estimated
using best approximation results. We shall now estimate ϑ, by quantities involving
the problem data and/or p, by using discrete stability estimates.
The model problem (2.10) in weak form with weakly imposed initial condition
reads: ﬁnd u ∈ H1
(I; V) such that
tn
0
(u , v)H + a(u, v) dt + (u(0), v(0))H =
tn
0
(f(u), v)H dt + (u0, v(0))H, (4.5)
for all v ∈ L2(I; V), and n = 1, . . . , N, which upon subtracting (2.23) summed for
j = 1, . . . , n, yields the identity
tn
0
((u − U) , v)H + a(u − U, v) dt −
n
j=2
([U]j−1, v+
j−1)H + ((u − U)+
0 , v+
0 )H
=
tn
0
(f(u) − Πf(U), v)H dt + (u0 − ˜Phu0, v+
0 )H
(4.6)
for all v ∈ Xr(Vh), n = 1, . . . , N. Upon setting v = ϑ ∈ Xr(Vh) in (4.6), gives
tn
0
(ϑ , ϑ)H + a(u − U, ϑ) dt −
n
j=2
([U]j−1, ϑ+
j−1)H + ϑ+
0
2
H
=
tn
0
(f(u) − Πf(U), ϑ)H dt −
tn
0
(p , ϑ)H dt − (p(0), ϑ+
0 )H.
(4.7)
Upon observing that −[U]j−1 = [ϑ]j−1 − [Pu]j−1 for j = 2, . . . , n, along with the
(classical) identity
tn
0
(w , w)H dt +
n
j=2
([w]j−1, w+
j−1)H + w+
0
2
H
=
1
2
w−
n
2
H +
1
2
n
j=2
[w]j−1
2
H +
1
2
w+
0
2
H,

(4.7) yields
1
2
ϑ−
n
2
H +
1
2
n
j=2
[ϑ]j−1
2
H +
1
2
ϑ+
0
2
H +
tn
0
a(u − U, ϑ) dt
=
tn
0
(f(u) − Πf(U), ϑ)H dt −
tn
0
(p , ϑ)H dt − (p(0), ϑ+
0 )H
+
n
j=2
([Pu]j−1, ϑ+
j−1)H
=
tn
0
(f(u) − Πf(U), ϑ)H dt +
tn
0
(p, ϑ )H dt − (p(tn), ϑ−
n )H
−
n−1
j=1
([p]j, ϑ+
j )H −
n−1
j=1
(p(t−
j ), ϑ−
j )H +
n−1
j=1
(p(t+
j ), ϑ+
j )H
=
tn
0
(f(u) − Πf(U), ϑ)H dt +
tn
0
(p, ϑ )H dt − (p(tn), ϑ−
n )H
+
n−1
j=1
(p(t−
j ), [ϑ]j)H,
(4.8)
by integration by parts with respect to the time variable and by noting that
p(t±
j ) = u(tj) − Pu(t±
j ).
Also, we have
a(u − U, ϑ) = a(u − Phu, ϑ) + a((I − πn
) ⊗ Phu, ϑ) + a(ϑ, ϑ), (4.9)
with I denoting the identity operator with respect to the t-variable in this partic-
ular instance. Upon invoking the defining property (3.6) of the elliptic projection,
the first term on the right–hand side of (4.9) vanishes and, thus, after integration
with respect to the time variable, we have
˜λ
tn
0
a(u − U, ϑ) dt =
tn
0
a((I − πn
) ⊗ Phu, ϑ) dt +
tn
0
a(ϑ, ϑ) dt. (4.10)
Again, the first term on the right-hand side of (4.10) vanishes from the orthogo-
nality of the piecewise L2-projection operator πn
with respect to the time variable

and the simplifying assumption (4.4); hence, (4.8) yields
1
2
ϑ−
n
2
H +
1
2
n
j=2
[ϑ]j−1
2
H +
1
2
ϑ+
0
2
H + Ccoer
tn
0
ϑ 2
V dt
≤
tn
0
(f(u) − Πf(U), ϑ)H dt +
tn
0
(p, ϑ )H dt − (p(tn), ϑ−
n )H
+
n−1
j=1
(p(t−
j ), [ϑ]j)H.
(4.11)
Using the coercivity of the elliptic operator. Standard arguments such as Cauchy-
Schwarz and Young inequalities now yield
1
2
ϑ−
n
2
H +
1
2
n
j=2
[ϑ]j−1
2
H +
1
2
ϑ+
0
2
H + Ccoer
tn
0
ϑ 2
V dt
≤
2
Ccoer
tn
0
f(u) − Πf(U) 2
V∗ dt +
Ccoer
8
tn
0
ϑ 2
V dt
+
tn
0
˜λ−1
p 2
H dt +
1
4
tn
0
˜λ ϑ 2
H dt + p(tn) 2
H +
1
4
ϑ−
n
2
H
+ 2
n−1
j=1
p(t−
j ) 2
H +
1
8
n−1
j=1
[ϑ]j
2
H.
(4.12)
For some ˜λ > 0 constant on each subinterval Ij to be deﬁned precisely below,
giving
1
4
ϑ−
n
2
H +
1
4
n
j=2
[ϑ]j−1
2
H +
1
2
ϑ+
0
2
H +
Ccoer
4
tn
0
ϑ 2
V dt
ˆλ ≤
2
Ccoer
tn
0
f(u) − Πf(U) 2
V∗ dt +
tn
0
˜λ−1
p 2
H dt
+
1
4
tn
0
˜λ ϑ 2
H dt + 2
n
j=1
p(t−
j ) 2
H.
(4.13)
We observe that the right–hand side of (4.13) includes ϑ which is not present on
the left–hand side. To deal with this term we employ the ideas from [27, 26], in
that we seek to strengthen the norm on the left–hand side of (4.12) via an inf-sup
condition argument. To that end, in line with the proof of [26, Theorem 4.5] (cf.
also, [82]), we set
v = ˜λϑ , where ˜λ|In := ˜γ
kn
r2
n
n = 1, . . . , N,

for some ˜γ > 0 constant (to be deﬁned precisely below) in (4.6), to arrive at
tn
0
˜λ ϑ 2
H + ˜λa(ϑ, ϑ ) dt −
n
j=2
˜λ([U]j−1, (ϑ )+
j−1)H + ˜λ(ϑ+
0 , (ϑ )+
0 )H
=
tn
0
˜λ(f(u) − Πf(U), ϑ )H dt −
tn
0
˜λ (p , ϑ )H + a(p, ϑ ) dt
− ˜λ(p(0), (ϑ )+
0 )H.
(4.14)
For t ∈ Ij, standard inverse estimates with respect to the time variable imply,
respectively,
˜λa(ϑ, ϑ ) ≤ ˜λCcont ϑ V ϑ V ≤ ˜γCCcont ϑ 2
V,
and
˜λ([U]j−1, (ϑ )+
j−1)H ≤ ˜λ [U]j−1 H (ϑ )+
j−1 H
≤ C˜λ
rj
kj
[U]j−1 H
Ij
ϑ 2
H dt
1
2
≤ C˜γ [U]j−1
2
H +
1
4 Ij
˜λ ϑ 2
H dt,
which, upon summation for j = 2, . . . , n gives
n
j=2
˜λ([U]j−1, (ϑ )+
j−1)H ≤ ˜λ [U]j−1 H (ϑ )+
j−1 H
≤ C˜λ
rj
kj
[U]j−1 H
Ij
ϑ 2
H dt
1
2
≤ C˜γ
n
j=2
[U]j−1
2
H +
1
4
tn
0
˜λ ϑ 2
H dt.
Similarly, for w ∈ {p(0), ϑ+
0 } ,we also have
˜λ(w, (ϑ )+
0 )H ≤ C˜λ
r1
√
k1
w H
I1
ϑ 2
H dt
1
2
≤ C˜γ w 2
H +
1
8 I1
˜λ ϑ 2
H dt.
Also, from (4.9) with ϑ replaced by ϑ ∈ X(Vh), we have
tn
0
˜λa(p, ϑ ) dt = 0,
since ˜λ is constant on each Ij.

Using the above estimates, along with standard arguments such as Cauchy-Schwarz
and Young inequalities into (4.14), we arrive at the bound
1
4
tn
0
˜λ ϑ 2
H dt ≤ C˜γ
n
j=2
[U]j−1
2
H + C˜γ ϑ+
0
2
H + C˜γ p(0) 2
H
+ CCcont˜γ
tn
0
ϑ 2
V dt
+
tn
0
2˜λ f(u) − Πf(U) 2
H + p 2
H dt.
(4.15)
Using (4.15) to bound the third term on the right-hand side of (4.13), along with
the bound
n
j=2
[U]j−1
2
H ≤
n
j=2
[ϑ]j−1
2
H +
n
j=2
[Pu]j−1
2
H,
(arising from the identity −[U]j−1 = [ϑ]j−1 − [Pu]j−1,) results in (4.13) giving
1
4
ϑ−
n
2
H +
1
8
n
j=2
[U]j−1
2
H +
1
2
ϑ+
0
2
H +
Ccoer
4
tn
0
ϑ 2
V dt
≤
2
Ccoer
tn
0
f(u) − Πf(U) 2
V∗ dt +
tn
0
˜λ−1
p 2
H dt
+ 2
n
j=1
p(t−
j ) 2
H + C˜γ
n
j=2
[U]j−1
2
H + C˜γ ϑ+
0
2
H + C˜γ p(0) 2
H
+ CCcont˜γ
tn
0
ϑ 2
V dt +
1
4
n
j=2
[Pu]j−1
2
H
+
tn
0
2˜λ f(u) − Πf(U) 2
H + p 2
H dt.
(4.16)
Upon selecting now ˜γ > 0 small enough so that C˜γ ≤ 1/32 and CCcont˜γ ≤
Ccoer/16, (4.16) ﬁnally implies
1
4
ϑ−
n
2
H +
1
16
n
j=2
[U]j−1
2
H +
1
16
ϑ+
0
2
H +
Ccoer
8
tn
0
ϑ 2
V dt
≤
2
Ccoer
tn
0
f(u) − Πf(U) 2
V∗ dt +
tn
0
˜λ−1
p 2
H dt
+ 2
n
j=1
p(t−
j ) 2
H + C˜γ p(0) 2
H +
1
4
n
j=2
[Pu]j−1
2
H
+
tn
0
2˜λ f(u) − Πf(U) 2
H + p 2
H dt.
(4.17)
To simplify matters, we postulate the validity of a Poincaré–Friedrichs inequality
between H and V; this is, of course, the case in the canonical pairs we have in

mind, such at H = L2(Ω) and V = H1
0 (Ω).
Assumption 4.3. There exists positive constant CPF , such that v 2
H ≤ CPF v 2
V
for all v ∈ V.
Hence, the above assumption leads to the inequality v 2
V∗ ≤ CPF v 2
H.
Using the last estimate, (4.17) then implies
ϑ−
n
2
H +
n
j=2
[U]j−1
2
H + ϑ+
0
2
H + Ccoer
tn
0
ϑ 2
V dt
≤ C
tn
0
f(u) − Πf(U) 2
H dt +
1
2
En(u)
≤ C
tn
0
Π(f(u) − f(U)) 2
H dt + En(u),
(4.18)
where
En(u) := C
tn
0
f(u) − Πf(u) 2
H dt +
tn
0
2˜λ−1
p 2
H + 4˜λ p 2
H dt
+ 4
n
j=1
p(t−
j ) 2
H + C˜γ p(0) 2
H +
1
2
n
j=2
[Pu]j−1
2
H.
Adding now four times (4.15) to (4.24) aiming to include the left–hand side of
(4.15) into the estimation and recalling that ˜γ is chosen small enough, we arrive
at
ϑ−
n
2
H +
1
4
n
j=2
[U]j−1
2
H +
1
4
ϑ+
0
2
H +
Ccoer
4
tn
0
ϑ 2
V dt +
tn
0
˜λ ϑ 2
H dt
≤ C
tn
0
Π(f(u) − f(U)) 2
H dt + 2En(u),
(4.19)
or, dropping the constants
ϑ−
n
2
H +
n
j=2
[U]j−1
2
H + ϑ+
0
2
H + Ccoer
tn
0
ϑ 2
V dt +
tn
0
˜λ ϑ 2
H dt
≤ C
tn
0
Π(f(u) − f(U)) 2
H dt + 8En(u).
(4.20)

4.2.2 Completing the bound
Now, [52, Lemma 4.3] ensures us that
f(u) − Πf(u) L∞(In;H) ≤ C max
n−−µ≤ ≤n−
min
s≤rn+1
ks
D(s)
f(u) L∞(In;H), (4.21)
i.e., we have optimal convergence with respect to the maximum timestep locally.
Also, recalling the uniform stability of the Lagrangian interpolation basis functions
used in the construction of Π from the proof of [52, Lemma 4.1], viz.,
|ξη(t)| ≤ C,
for C independent of the local timestep (the validity of this estimate can be shown
upon observing that the support of ξη(t) grows proportionally with the polynomial
degree), we deduce
tn
0
Π(f(u) − f(U)) 2
H dt ≤ C
n
m=1
m−
η=m−−µ
max
n−−µ≤η≤n−
kη f(u(tη)) − f(Uη) 2
H.
(4.22)
Remark 4.4. Despite our eﬀort in being explicit with respect to the local polyno-
mial degree in the time variable in this a priori error analysis, we are not aware
of the mode of dependence of the constants C in (4.21) and (4.22). We do expect,
however, that they decrease as the local polynomial degree increases.
Now, upon identifying f : R → R with a function f : H → H by f(v(t, x)) :=
(f(v(t)))(x) with x being the spatial variable, we also consider fL
: H → H
satisfying
fL
(w) − fL
(v) H ≤ CL w − v H, (4.23)
i.e., a globally Lipschitz function, such that we have f(v) = fL
(v), for all v ∈ H
with v H ≤ L := 2 max0≤t≤T u(t) H. This implies, in particular, that fL
(u) =
f(u). Upon replacing f by fL
on the numerical method (2.23), we denote the
resulting numerical solution by UL
∈ Xr(Vh). Therefore, (4.24) and (4.22) hold

with U replaced by UL
and f(U) replaced by fL
(UL
), giving
(ϑL
)−
n
2
H +
n
j=2
[UL
]j−1
2
H + (ϑL
)+
0
2
H +
tn
0
Ccoer ϑL 2
V + ˜λ (ϑL
) 2
H dt
≤ C
n
m=1
˜km f(u(tm)) − fL
(UL
m) 2
H + 8En(u),
(4.24)
where we have introduced the notation ˜km := µ maxn−−µ≤η≤n− kη and ϑL
:=
Pu − UL
. Due to (4.23), the ﬁrst term on the right–hand side of (4.24) can,
therefore, be further estimated as follows:
f(u(tm)) − fL
(UL
m) H ≤ CL u(tm) − UL
m H ≤ CL p(tm) H + CL ϑL
m H,
which, in conjunction with (4.24) yields
(ϑL
)−
n
2
H +
n
j=2
[UL
]j−1
2
H + (ϑL
)+
0
2
H +
tn
0
Ccoer ϑL 2
V + ˜λ (ϑL
) 2
H dt
≤ CCL
n
m=1
˜km ϑL
m
2
H, +2CL
n
m=1
p(tm) 2
H + 8En(u).
(4.25)
This, upon further assuming that there exists a constant cquas > 0 such that
˜km ≤ cquas min{rm, m}km, for all m ∈ {1, . . . , N}, (4.26)
uniformly, in conjunction with the discrete version of the Grönwall inequality, gives
(ϑL
)−
n
2
H +
n
j=2
[UL
]j−1
2
H + (ϑL
)+
0
2
H +
tn
0
Ccoer ϑL 2
V + ˜λ (ϑL
) 2
H dt
≤ exp(CCLrmax) 2CL
n
m=1
p(tm) 2
H + 8En(u) =: Emax
n (u),
(4.27)
with rmax := max{max1≤n≤N rn, N}.
Now, using standard approximation estimates (hp–version approximation esti-
mates) we can see that the right-hand side of (4.27) decays to zero, as the maxi-
mum timestep and the maximum diameter of the spatial elements converge to zero
and/or as the respective temporal and spatial polynomial degrees in the space-time
method increase. Assuming, however, for the moment that this is, indeed, the case,

we aim to prove that UL
= U. Hence, we want to show that
max
0≤t≤T
UL
(t) H ≤ L, (4.28)
because this would mean that fL
(UL
) = f(UL
) and, thus (4.22) is valid with
f (as per original method), which necessarily implies that UL
= U, since they
are solutions to the same method. Implicitly, the last statement assumes the
uniqueness of the solution of the numerical method (4.22), which we shall assume
in the final theorem.
To this end, we have
max
0≤t≤T
UL
(t) H ≤ max
0≤t≤T
u − UL
(t) H + max
0≤t≤T
u H = max
0≤t≤T
u − UL
(t) H +
L
2
.
Therefore, it is enough to prove that max0≤t≤T u − UL
(t) H ≤ L/2 also. To do
so, we employ the triangle inequality and Lemma 4.1 as follows:
max
0≤t≤T
u − UL
(t) H ≤ max
0≤t≤T
u − Pu H + max
0≤t≤T
ϑL
(t) H
≤ max
0≤t≤T
u − Pu H + max
t∈Ij∗
ϑL
(t) H
≤ max
0≤t≤T
u − Pu H + C kj∗
Ij∗
(ϑL
) 2
H dt
+ ϑ−
j∗
2
H
1
2
,
(4.29)
for j∗
the index of an interval Ij∗ on which the maximum is attained. Therefore,
Assumption 4.3 and (4.27) finally give
max
0≤t≤T
u − UL
(t) H ≤ max
0≤t≤T
u − Pu H + rmax Emax
N (u)
1
2
, (4.30)
with tN = T, i.e., the final time. Since the right–hand side of (4.30) can be chosen
arbitrarily small by selecting sufficiently small maximum time–steps and spatial
meshsizes and/or sufficiently large polynomial degrees with respect to the time and
the space discretisations, we can conclude that, for such discretisation parameters
the right–hand side of (4.30) is less than or equal to L/2. This, as discussed above,
in turn yields that (4.28) holds and, therefore, UL
= U. Hence, (4.27) holds with

UL
= U and, thus, with ϑL
= ϑ, viz.,
ϑ−
n
2
H +
n
j=2
[U]j−1
2
H + ϑ+
0
2
H +
tn
0
Ccoer ϑ 2
V + ˜λ ϑ 2
H dt ≤ Emax
n (u). (4.31)
Therefore, we have already proven the following result.
Theorem 4.5. With the above assumptions, for sufficiently small spatial and tem-
poral meshsizes and/or sufficiently large polynomial degrees so that
max
0≤t≤T
u − Pu H + rmax Emax
n (u)
1
2
≤ u L∞(0,tn;H),
the following bounds hold
u(tn) − U−
n
2
H +
n
j=2
[U]j−1
2
H + u(0) − U+
0
2
H
+ Ccoer
tn
0
u − U 2
V dt + ˜λ (u − U) 2
H dt
≤ Emax
n (u) + Ccoer
tn
0
u − Pu 2
V dt,
(4.32)
and
u(tn)−U−
n
2
H+
n
j=2
[U]j−1
2
H+ u(0)−U+
0
2
H+˜λ (u−U) 2
H dt ≤ Emax
n (u). (4.33)
Proof. The proof follows immediately by the triangle inequality.
Going back to the growth assumption (2.14) for the nonlinear reaction f, upon
assuming that both u and U are bounded in L∞(I; V), with the latter indepen-
dent from the mesh parameters, we can conclude that f satisfies a local Lipschitz
condition of the form
f(u) − f(U) H ≤ C(u, U) u − U H,
for which we can conclude (4.23) needed for the proof of the above a priori bounds.
We finally remark on the optimality of the above a priori error bounds. The use
of the elliptic projection in conjunction with the L2-projection in the time variable

will lead to optimal a priori error bounds in the L2(H1
)-norm. As we shall see,
however, the respective a priori bounds in the L∞(L2)–norm error are slightly
suboptimal by half an order of kn, due to the presence of the term
n
j=1
p(t−
j ) 2
H
in Emax
n (u). We shall comment further on this point further below.
We are now in a position to ﬁnalise the a priori error analysis.
4.2.3 A priori error bounds
We are now ready to complete the a priori error analysis.
Theorem 4.6 (L∞(I; H)–norm estimate). Assuming the validity of estimate (4.22)
and of Assumption 4.2, (or, in the special case of H = L2(Ω), assuming the hy-
potheses of Theorem 4.5 and Lemma 4.1, respectively) and assuming the regularity
u(η)
|In ∈ L2(In; H) and u|In ∈ Hκn
(In; H) for some η ≥ 2 and κn ≥ 2, for each
n = 1, . . . , N. Then, for n = 1, ..., N, we have the a priori error bound
u − U 2
L∞(0,tn;H) ≤ C
n
j=1
k
2sj+1
j
r
2sj
j
u(sj) 2
L2(Ij;H) +
r2
j
kj
h2t+2 (t)
u 2
L2(Ij;H) ,
(4.34)
for every 1 ≤ sj ≤ min{rj, κj} and 1 ≤ t ≤ min{rs, η}, where rs denotes the
polynomial degree of the space discretisation.
Proof. In view of Assumption 4.2 (or of Lemma 4.1), along with (4.33), we have
u − U 2
L∞(In;H) ≤ C kn
In
(u − U) (t) 2
H dt + (u − U)(t−
n ) 2
H
≤ CrmaxEmax
n (u).
(4.35)
We now estimate the right–hand side of the last bound via the use of standard
hp–version approximation results. From hp-version approximation estimates for

the L2-projection πn
, see, e.g., [? ], we have on each In:
In
˜λ−1
p 2
H dt ≤ 2
In
˜λ−1
u − πn
u 2
H dt +
In
˜λ−1
πn
(u − Phu) 2
H dt
≤ C
k2s+1
n
r2s
n
u(s) 2
L2(In;H) + C
r2
n
kn
h2t+2 (t)
u 2
L2(In;H),
for some 0 ≤ s ≤ min{rn, κ} and 0 ≤ t ≤ min{rs, η}, where rs denotes the
polynomial degree of the space discretisation. Working analogously and using a
standard inverse estimate, we also have
In
˜λ p 2
H dt ≤ 2
In
˜λ (u − πn
u) 2
H dt + 2
In
˜λ (πn
(u − Phu)) 2
H dt
≤ 2
In
˜λ (u − πn
u) 2
H dt + C
In
πn
(u − Phu) 2
H dt
≤ C
k2s+1
n
r2s+2
n
u(s) 2
L2(In;H) + Ch2t+2 (t)
u 2
L2(In;H),
for 1 ≤ s ≤ min{rn, κ} and 0 ≤ t ≤ min{rs, η}. Further, using the trace–inverse
estimate, and approximation estimates from the boundary to In, see, e.g., [56], we
have
p(t−
n ) 2
H ≤ 2 u(tn) − πn
u(t−
n ) 2
H + 2 πn
(u − Phu)(t−
n )) 2
H
≤ 2 u(tn) − πn
u(t−
n ) 2
H + C
r2
n
kn In
πn
(u − Phu)) 2
H dt
≤ 2 u(tn) − πn
u(t−
n ) 2
H + C
r2
n
kn In
u − Phu 2
H dt
≤ C
k2s+1
n
r2s
n
u(s) 2
L2(In;H) + C
r2
n
kn
h2t+2 (t)
u 2
L2(In;H),
and, completely analogously for p(0) 2
H, giving
p(0) 2
H ≤ C
k2s+1
1
r2s
1
u(s) 2
L2(I1;H) + C
r2
1
kn
h2t+2 (t)
u 2
L2(I1;H).

Finally, the trace inequality and working as above implies
n
j=2
[Pu]j−1
2
H = [u − Pu]j−1
2
H
≤ 2
n
j=2
(u − Pu)(t−
j−1) 2
H + (u − Pu)(t+
j−1) 2
H
= 2
n
j=2
p(t−
j−1) 2
H + p(t+
j−1) 2
H
≤ C
n
j=1
k2s+1
j
r2s
j
u(s) 2
L2(Ij;H) +
r2
j
kj
h2t+2 (t)
u 2
L2(Ij;H) .
Combining the above, the result already follows.
Similarly, we have an a priori bound in the L2(I, V)–norm.
Theorem 4.7 (L2(I; V)–norm estimate). Assuming the validity of estimate (4.22),
(or, in the special case of H = L2(Ω), assuming the hypotheses of Theorem 4.5),
and assuming the regularity (η)
u|In ∈ L2(In; H), (η−1)
u|In ∈ L2(In; V), u|In ∈
Hκn
(In; H), and u|In ∈ Hκn−1
(In; V), for some η ≥ 2 and κn ≥ 2, for each
n = 1, . . . , N. Then, for n = 1, ..., N, we have the a priori error bound
u − U 2
L2(0,tn;V) ≤ C
n
j=1
k
2sj+1
j
r
2sj
j
u(sj) 2
L2(Ij;H) +
r2
j
kj
h2t+2 (t)
u 2
L2(Ij;H) ,
(4.36)
for every 1 ≤ sj ≤ min{rj, κj} and 1 ≤ t ≤ min{rs, η}.
Proof. The proof follows as the respective one in the previous theorem with the
addition of estimating the term
In
u − Pu 2
V dt ≤ 2
In
u − πn
u 2
V dt + 2
In
πn
(u − Phu) 2
V dt
≤
k2sn
n
r2sn
n
u(sn−1) 2
L2(In;V) + 2
In
πn
(u − Phu) 2
V dt
≤
k2sn
n
r2sn
n
u(sn−1) 2
L2(In;V) + Ch2t (t−1)
u 2
L2(In;V),
and the proof already follows.
We remark that the bound in Theorem 4.6 is slightly suboptimal by half an order
of kn with respect to the time discretisation. It is possible to use duality arguments

to recover optimal rate for the case of linear problems [109]. However, this has not
been possible to extend in the current nonlinear setting of only locally Lipschitz
continuous nonlinearities. Instead, we opted for the “inf-sup”–type argument from
[26, 27] which is more general but delivers this slightly suboptimal rate.
4.3 Numerical examples
We present a series of numerical experiments to study the asymptotic convergence
behaviour of the dG time–stepping methods with continuous finite elements in
space i.e. dG(r)–cG(p). We report the experimental order of convergence (EOC)
relative to the last computed quantities in all figures as an indication of the asymp-
totic rate of convergence. In all cases, A = ∆, i.e., the Dirichlet Laplacian, yield-
ing the heat equation with linear source term and H = L2(Ω), V = H1
0 (Ω), giving
H∗
= H−1
(Ω). The numerical implementation is based on the deal.II finite el-
ement library [16] and the tests run in the high performance computing facility
ALICE at the University of Leicester.
4.3.1 Example 1
We consider the heat equation as a standard example of the linear parabolic prob-
lems, where the initial condition and the right hand side function are chosen such
that the exact solution is
u(x, y, t) = e−t
x(1 − x)y(1 − y).
We solve the problem on the space–time cylinder I × Ω := [0, 1] × [0, 1]2
, on a
fixed uniform rectangular mesh consisting of 1024 uniform biquadratic elements
in space (p = 2), with elements of orders r = 0, 1, 2, 3, 4 in time. We study the
asymptotic behaviour of the error e in L2(H1
)-, L2(L2)-, L∞(L2)-, L∞(L∞)-, and
L∞(H1
)-error norms and also we examine the superconvergence of the ∞(L2)-
error norm at the endpoints of the time intervals by monitoring the evolution of

the experimental order of convergence (EOC) over time on a sequence of uniformly
refined meshes in time. In each instance, we fix a constant mesh step size h = 1/32
and we also use fixed polynomial degree in space with various polynomial degrees
in time (dG(r)-cG(2)), r = 0, 1, 2, 3, 4. The resulting errors are plotted against
the corresponding time step size kn. In the Figure 4.1 (a)–(e) below, we notice
the optimal order of convergence of the L2(H1
)-, L2(L2)-, L∞(L2)-, L∞(L∞)-,
and L∞(H1
)-error norms, respectively, which is r + 1 of the polynomial degrees
r = 0, 1, 2, 3, 4. Figure 4.1 (f) shows the superconvergence of the ∞(L2)-error
norm at the endpoints of the time intervals. The superconvergence is investigated
to show that the method has better convergence properties at the time interval
endpoints than within the time interval. The results confirm the theoretical results
of Theorems 4.6 and 4.7.
4.3.2 Example 2
We solve in this example the same problem as in Example 4.3.1 on the space–
time cylinder I × Ω := [0, 0.1] × [0, 1]2
, on a fixed uniform rectangular mesh
consisting of 1024 uniform quartic elements in space (p = 4), with elements of
orders r = 0, 1, 2, 3, 4 in time. We study the asymptotic behaviour of the error e
in L2(H1
)-, L2(L2)-, L∞(L2)-, L∞(L∞)-, and L∞(H1
)-error norms and, also, we
examine the superconvergence of the ∞(L2)-error norm at the endpoints of the
time intervals by monitoring the evolution of the experimental order of convergence
(EOC) over time on a sequence of uniformly refined meshes in time.
In each instance, we fix a constant mesh step size h = 1/32 and we also use fixed
polynomial degree in space with various polynomial degrees in time (dG(r)-cG(4)),
r = 0, 1, 2, 3, 4. The resulting errors are plotted against the corresponding time
step size kn. In the figure (a) below, we notice that all the error norms mentioned
above have linear convergence (dG(0)-cG(4), also, we observe that there is no
superconvergence in this case (where SCon stands for superconvergence) since
dG(0) is equivalent to the backward Euler method. The Figure 4.2 (b)-(e) for the
cases dG(r)-cG(4), r = 1, 2, 3, 4, respectively, show that the error norms mentioned

(a) (b)
(c) (d)
(e) (f)
Figure 4.1: Example 1: h–version IMEX dG(r)–cG(2) scheme, r = 0, 1, 2, 3, 4,
for diﬀerent error norms vs the time steps kn.

above have optimal order of convergence EOC ≈ r + 1 and superconvergence of
the ∞(L2)-error norm with EOC ≈ r + 2. The results are in agreement with the
theoretical results of Theorems 4.6 and 4.7.
4.3.3 Example 3
We solve the same problem as in Example 4.3.1. We consider in this example the
p–version IMEX dG time–advancing schemes. We solve the problem on I × Ω :=
[0, 1]×[0, 1]2
on a fixed uniform rectangular mesh consisting of 1024 uniform quartic
elements in space (p = 4), and different time elements of orders r = 0, 1, 2, 3, 4
with fixed time step size kn = 0.01 and space mesh h = 1/16.
For the p–version, Figure 4.3 shows the error for the numerical method in the
L2(H1
)-, L2(L2)-, L∞(L2)-, and L∞(L∞)-error norms for fixed space–time mesh
size under p–refinement. We observe exponential convergence in these error norms
since the solution is analytic over the computational domain.
4.3.4 Example 4
We implement in this Example the h–version IMEX dG time—marching schemes
of the heat equation with the initial condition and source function are chosen such
that the exact solution is
u(x, y, t) = e−t
sin(πx) sin(πy),
on the space–time cylinder I ×Ω := [0, 0.1]×[0, 1]2
, on a fixed uniform rectangular
mesh consisting of 1024 uniform quintic elements in space (p = 5), with uniform
quadratic elements in time r = 2. We study the asymptotic behaviour of the error
e in L2(H1
)-, L2(L2)-, L∞(L2)-, L∞(L∞)-, and L∞(H1
)-error norms and also we
we examine the superconvergence of the ∞(L2)-error norm at the endpoints of the
time intervals by monitoring the evolution of the experimental order of convergence
(EOC) over time on a sequence of uniformly refined meshes in time.

(a) (b)
(c) (d)
(e)
Figure 4.2: Example 2: h–version IMEX dG(r)–cG(4), r = 0, 1, 2, 3, 4 for
diﬀerent error norms vs the time steps kn.

Figure 4.3: Example 3: p–version IMEX dG timestepping scheme for r = 2
and time step kn = 0.01, for diﬀerent error norms.

In each instance, we fix a constant mesh step size h = 1/32 and we also use fixed
polynomial degrees in both space and time (dG(2)-cG(5)). In the Fig. 4.4 below,
we notice that all the error norms mentioned above have cubic convergence, also,
we observe the superconvergence in the ∞(L2)-error norm EOC ≈ 4. Note that
N.SDof it means the total number of space degrees of freedom. The numerical
results coincide with the theoretical results of Theorems 4.6 and 4.7.
4.3.5 Example 5
We implement in this Example the h–version IMEX dG time—marching scheme
with the initial condition and source function are chosen such that the exact so-
lution is
u(x, y, t) = tα
x(1 − x)y(1 − y).
We solve the problem over the computational domain I ×Ω := [0, 0.1]×[0, 1]2
, on
a fixed uniform rectangular mesh consisting of 1024 uniform quintic elements in
space p = 5 and uniform quadratic elements in time r = 2, with fixed mesh size
h = 1/32, over a sequence of algebraically graded meshes in time with grading
factor α = 0.75.
This solution has initial layer and low regularity at t = 0 but it is analytic over the
spatial domain Ω. We use temporal meshes, geometrically graded towards t = 0, to
achieve exponential rates of convergence. For this reason, we consider a short time
interval with T = 0.1. Let 0 < ˜λ < 1 be the mesh grading factor which defines a
class of temporal meshes tn = ˜λN−n
, n = 1, ..., N. In this example, we set ˜λ = 0.5.
The Fig. 4.5 shows that the convergence rates are recovered by using algebraically
graded meshes in the L2(H1
)-, L2(L2)-, L∞(L2)-, L∞(L∞)-, and L∞(H1
)-error
norms with the expected EOC ≈ 2, and also the nodal superconvergence in the
∞(L2) norm with EOC ≈ 3.

Figure 4.4: Example 4: h–version IMEX dG timestepping dG(2)–cG(5)
scheme for diﬀerent error norms.

Figure 4.5: Example 5: h–version on algebraically graded meshes dG(2)–
cG(5) for diﬀerent error norms.

Chapter 5
Conclusions
5.1 Conclusions
In this work we studied discontinuous Galerkin timestepping for semilinear parabolic
problems. In particular, we considered fully discrete implicit–explicit (IMEX) vari-
ational discretisations using the discontinuous Galerkin (dG) method in time com-
bined with standard (continuous) Galerkin (cG) finite element methods in space.
The time discretisation consists of a hp–version discontinuous Galerkin method
treating implicitly the diffusion spatial operator and using an explicit multistep
method for the nonlinear reaction term. We analysed general dG(r)–cG(p) combi-
nations, where r is the polynomial degree in time and p is the polynomial degree in
space. These methods were first proposed and analysed in the a priori setting by
Estep and Larsson [52] under the assumption of globally Lipschitz nonlinearities.
We derived optimal L∞(L2) and L2(H1
) a posteriori error bounds under the more
general assumption of locally Lipschitz continuous nonlinearities satisfying a cer-
tain growth condition dictated by suitable Sobolev imbedding results. The analysis
builds on new a posteriori error estimates for linear parabolic problems presented
in [60], using the elliptic reconstruction technique of Makridakis and Nochetto [83].
The performance of the error estimators are highlighted by a set of numerical ex-
amples, confirming that the a posteriori error estimators are optimal, reliable, and
efficient.
85

Conclusions 86
We also consider the challenging problem of extending the a priori error analy-
sis of discontinuous Galerkin timestepping methods to semilinear problems with
merely locally-Lipschitz continuous nonlinear reaction terms. In this setting, we
derived a priori error bounds in the L∞(L2) and L2(H1
) norms. The analysis is
based on the classical elliptic projection technique and discrete stability estimates
combined with an inf-sup argument in time. A ﬁxed-point argument combined
with a discrete version of the Grönwall inequality is used to control the nonlinear
terms in the spirit of [4, 29]. The treatment of general nonlinearities comes at
the expense of certain assumptions, such as local quasi-uniformity of the timestep
and boundedness of the exact and approximate solutions. By using hp-version ap-
proximation estimates we were able to derive the analysis keeping the dependence
on the polynomial degree as much as possible explicit. Furthermore, we tested
the a priori error estimates by implementing a series of numerical examples. The
results of the numerical experiments are in agreement with the theoretical results
and, in the particular case of the L∞(L2)–error norm, the observed behaviour is
better than what is proven by about half an order.
An interesting aspect of the a posteriori analysis concerning implicit–explicit time
stepping methods, is that no a priori CFL type conditions are required for the
validity of the conditional a posteriori error bounds. Hence, the a posteriori esti-
mators remain reliable even for unstable combinations of local spatial and temporal
mesh sizes. In future work, we will consider using this property to estimate CFL
constants in a rigorous, a posteriori fashion.
The study of nonlinear time–dependent PDE problems necessitates further in-
vestigation, as a number of important issues are yet to be addressed. One of
these issues-, is the derivation of a posteriori error estimates for explicit and
implicit–explicit timestepping methods for evolution PDEs, especially treating
fully–discrete numerical schemes. There is a very limited number of works dis-
cussing a posteriori error bounds for explicit timestepping methods for linear evolu-
tion problems [58, 57]. The challenge of studying the explicit (or implicit–explicit)
timestepping schemes in the context of rigorous a posteriori error control is the
careful construction of an implicit perturbation of the explicit scheme for which

Conclusions 87
we can construct suitable, optimal order, reconstructions that, in turn, can be
naturally inserted into the original PDE to construct residuals.
Regarding the a priori analysis, the study of semilinear evolution problems is
still a challenge, since the classical timestepping typically are deﬁned only on
time–nodes. In the discontinuous Galerkin timestepping schemes however, the
approximate solution is available on the whole time interval but it is discontinuous
at the time–nodes and a careful analysis is needed in this case. In the future, we
aim to apply the techniques we used in the a posteriori error analysis, namely,
the dG reconstruction technique [84] combined with the continuous version of the
Grönwall inequality, to derive optimal a priori error estimates for the semilinear
parabolic problems.

Appendix A
Numerical computations of
Chapter 2
A.1 Matrix form of the dG–timestepping schemes
for semilinear parabolic problems
The matrix form representation of the fully space–time discrete scheme in (2.46)
for the problem in (2.10) is given by
88

Appendix A Appendix A 89














0,0M + knβ0,0S · · · 0,rM + knβ0,rS
1,0M + knβ1,0S · · · 1,rM + knβ1,rS
...
...
...
r−1,0M + knβr−1,0S · · · r−1,rM + knβr−1,rS
r,0M + knβr,0S · · · r,rM + knβr,rS




























U0
n
U1
n
...
Ur−1
n
Ur
n














=














σ0MU
(0)
n−1
σ1MU
(0)
n−1
...
σr−1MU
(0)
n−1
σrMU
(0)
n−1














+
(A.1)














kn 0,n−−µM kn 0,n−−(µ−1)M · · · kn 0,n−−(µ−(µ−1))M kn 0,n−M
kn 1,n−−µM kn 1,n−−(µ−1)M · · · kn 1,n−−(µ−(µ−1))M kn 1,n−M
...
...
...
...
...
kn r−1,n−−µM kn r−1,n−−(µ−1)M · · · kn r−1,n−−(µ−(µ−1))M kn r−1,n−M
kn r,n−−µM kn r,n−−(µ−1)M · · · kn r,n−−(µ−(µ−1))M kn r,n−M




























f(U−
n−−µ)
f(U−
n−−(µ−1))
...
f(U−
n−−(µ−(µ−1)))
f(U−
n−)














.
When  = 0 we have the fully implicit timestepping scheme and when  = 1 we
obtain the implicit–explicit (IMEX) timestepping scheme.

Similarly, the matrix form representation for the fully space–time discrete scheme
of the system of semilinear parabolic equations in (2.48) is given by

























0,0M + l1knβ0,0S 0,1M + l1knβ0,1S · · · 0,rM + l1knβ0,rS
...
...
...
...
r,0M + l1knβr,0S r,1M + l1knβr,1S · · · r,rM + l1knβr,rS
...
...
...
...
r,0M + l2knβr,0S r,1M + l2knβr,1S · · · r,rM + l2knβr,rS


















































U0
n
U1
n
...
Ur
n
V0
n
V1
n
...
Vr
n

























=

























σ0MU
(0)
n−1
σ1MU
(0)
n−1
...
σrMU
(0)
n−1
σ0MV
(0)
n−1
σ1MV
(0)
n−1
...
σrMV
(0)
n−1

























(A.2)
+

























kn 0,n−−µM kn 0,n−−(µ−1)M · · · kn 0,n−M
kn 1,n−−µM kn 1,n−−(µ−1)M · · · kn 1,n−M
...
kn r,n−−µM kn r,n−−(µ−1)M · · · kn r,n−M
kn 0,n−−µM kn 0,n−−(µ−1)M · · · kn 0,n−M
kn 1,n−−µM kn 1,n−−(µ−1)M · · · kn 1,n−M
...
kn r,n−−µM kn r,n−−(µ−1)M · · · kn r,n−M


















































f(U−
n−−µ, V−
n−−µ)
f(U−
n−−µ−1, V−
n−−µ−1)
...
f(U−
n−, V−
n−)
g(U−
n−−µ, V−
n−−µ)
g(U−
n−−µ−1, V−
n−−µ−1)
...
g(U−
n−, V−
n−)

























.
Also, when  = 0 we have the fully implicit timestepping scheme and when  = 1
we obtain the implicit–explicit (IMEX) timestepping scheme.
A.2 Starting process on the previous time inter-
vals
As we mentioned in Chapter 2, we use a multistep interpolation process to ap-
proximate the nonlinear term on the right–hand side of our semilinear problems,
whether it is a single equation or a system. Hence, to evaluate the method on
the current time interval In we need the solution values on previous time intervals

and/or the current time interval. We will give below a detailed explanation of how
to start and proceed with our time marching schemes.
A.2.1 Starting process when  = 0 (The implicit case)
Assume that the order of the method in time is r. Ideally, the interpolant of the
nonlinear source term Πf(U) should be taken of order µ = 2r. In the solution pro-
cess, in order to be able to solve the nonlinear problem on the time interval Iµ, the
interpolant values on the previous time intervals I1, I2, ..., Iµ−1 are required, and
also on the current time interval Iµ i.e. we need Πf(U−
n ), n = 1, ..., µ. Since the in-
terpolant Πf(U) on the time interval Iµ is of order µ then we need µ+1 time nodes
tµ, tµ−1, tµ−2, ..., t1, t0 to construct this interpolating polynomial and then we need
the solution values at these support time points U−
µ , U−
µ−1, U−
µ−2, ..., U−
1 , U−
0 . We
can compute the interpolant values at these time points and solutions values, via
computing the source term values at these time nodes and solution values i.e.
f(U−
µ ), f(U−
µ−1), f(U−
µ−2), ..., f(U−
1 ), f(U−
0 ). Hence in this case we need the first
µth time intervals to construct this polynomial interpolant of order µ.
However, this is not possible on the first (µ − 1) time intervals. The interpolant
Πf(U) is of order µ on the intervals starting from the interval Iµ onwards i.e. for
the intervals Iµ, Iµ+1, ..., IN . For the remaining intervals I1, I2, ..., Iµ−1, the
interpolant has to be different on each interval. The interpolant on the interval
Iµ−1 is of order µ − 1 and on the interval Iµ−2 it is of order µ − 2 and so on until
the interval I1 where the interpolant is linear.
In summary, if the order of time polynomial is r (i.e. when using the dG(r) time
stepping scheme) then we need the interpolant of the nonlinear source term Πf(U)
to be of degree µ = 2r. To determine the degree of the source term interpolant
Πf(U) on any time interval In we have to cases:
(1) For the first µ time intervals In, n = 1, ..., µ the degree of the source term
interpolant Πf(U) is the same as the index of the time interval In i.e. µ = n.
Then the interpolant on the first time interval I1 is linear, µ = 1, and on the

second time interval I2 is quadratic, µ = 2, and so on until the µth time interval
Iµ where the interpolant degree is µ.
(2) The interpolant degree Πf(U) on all the remaining intervals Iµ+1, Iµ+2, ..., IN
is µ i.e the same degree of interpolant on the interval Iµ.
(I) The solution on the ﬁrst time interval I1 when n = 1.
To proceed with the solution process, we start from the ﬁrst time interval I1. On
this interval we need to construct the linear interpolant Π1
1f(U) = f(U−
0 )ξ0(t) +
f(U−
1 )ξ1(t). Hence we need only to compute f(U−
1 ), since (f(U−
0 ) is known from
the initial value) i.e. we need to solve the nonlinear system on this interval to
obtain the solution nodal values vector U−
1 at the time node t1, which we will
need for computing the interpolant of the right hand side to solve on the next
time interval I2, and so on. Now we can solve the problem (2.46) to obtain the
following nonlinear system










0,0M + k1β0,0S 0,1M + k1β0,1S · · · 0,rM + k1β0,rS
1,0M + k1β1,0S 1,1M + k1β1,1S · · · 1,rM + k1β1,rS
...
...
...
...
r,0M + k1βr,0S r,1M + k1βr,1S · · · r,rM + k1βr,rS




















U0
1
U1
1
...
Ur
1










=










σ0MU
(0)
0
σ1MU
(0)
0
...
σrMU
(0)
0










+










k1 0,0M k1 0,1M
k1 1,0M k1 1,1M
...
k1 r,0M k1 r,1M













f(U−
0 )
f(U−
1 )


 . (A.3)
Note that, here, U
(0)
0 and U−
0 actually represent the same function, that is, the
known solution at t = 0 while U−
1 = [U0
1, U1
1, . . . , Ur
1] is the unknown solution at
t = t1.
(II) The Solution on the second time interval I2 when n = 2.
We proceed now to the second interval I2 and the interpolant is taken to be a

quadratic polynomial
(Πf(U), v)H = (Π2
2f(U), v)H on I2,
where
Π2
2f(U) = f(U−
0 )ξ0(t) + f(U−
1 )ξ1(t) + f(U−
2 )ξ2(t),
by solving the following problem
I2
((U , v)H + a(U, v)) dt + ([U]1, v+
1 )H =
I2
(Π2
2f(U), v)H dt, ∀v ∈ V2. (A.4)
we get the following linear system










0,0M + k2β0,0S 0,1M + k2β0,1S · · · 0,rM + k2β0,rS
1,0M + k2β1,0S 1,1M + k2β1,1S · · · 1,rM + k2β1,rS
...
...
...
...




















U0
2
U1
2
...
Ur
2










=










σ0MU
(0)
1
σ1MU
(0)
1
...
σrMU
(0)
1










+










k2 0,0M k2 0,1M k2 0,2M
k2 1,0M k2 1,1M k2 1,2M
...
...
...
k2 r,0M k2 r,1M k2 r,2M

















f(U−
0 )
f(U−
1 )
f(U−
2 )







. (A.5)
(III) The solution on the µth time interval Iµ when n = µ.
Now, we can solve on the µth interval Iµ by using the µth degree interpolant
(Πf(U), v)H = (Πµ
µf(U), v)H on Iµ,
where
Πµ
µf(U) = f(U−
0 )ξ0(t) + f(U−
1 )ξ1(t) + ... + f(U−
µ )ξµ(t),

we solve now the following problem on the interval µ
Iµ
((U , v)H + a(U, v)) dt + ([U]µ−1, v+
µ−1)H =
Iµ
(Πµ
µf(U), ν)H dt, ∀v ∈ Vµ.
(A.6)
Finally, we have the following linear system














0,0M + knβ0,0S · · · 0,rM + knβ0,rS
1,0M + knβ1,0S · · · 1,rM + knβ1,rS
...
...
...
r−1,0M + knβr−1,0S · · · r−1,rM + knβr−1,rS
r,0M + knβr,0S · · · r,rM + knβr,rS




























U0
n
U1
n
...
Ur−1
n
Ur
n














=














σ0MU
(0)
n−1
σ1MU
(0)
n−1
...
σr−1MU
(0)
n−1
σrMU
(0)
n−1














+














kn 0,n−µM kn 0,n−(µ−1)M · · · kn 0,n−(µ−(µ−1))M kn 0,nM
kn 1,n−µM kn 1,n−(µ−1)M · · · kn 1,n−(µ−(µ−1))M kn 1,nM
...
...
...
...
...
kn r−1,n−µM kn r−1,n−(µ−1)M · · · kn r−1,n−(µ−(µ−1))M kn r−1,nM
kn r,n−µM kn r,n−(µ−1)M · · · kn r,n−(µ−(µ−1))M kn r,nM




























f(U−
n−µ)
f(U−
n−(µ−1))
...
f(U−
n−(µ−(µ−1)))
f(U−
n )














.(A.7)
A.2.2 Starting process when  = 1 (The implicit–explicit
case)
As we mentioned before, the interpolant of the nonlinear source term Πf(U) on
the first µth intervals is different from the interpolant on the interval Iµ+1 onwards.
The interpolant Πf(U) is of order µ on the intervals starting from the interval Iµ+1
onwards i.e. for the intervals Iµ+1, Iµ+2, ..., IN . For the remaining intervals I1,
I2, ..., Iµ−1, Iµ, the interpolant will be different and its order on each interval is
1, 2, · · · , µ−1, µ, except for the first interval where a predictor-corrector procedure
based on a constant and linear interpolant is used. For brevity, we will not repeat
the same details since most of them are similar to the implicit case. To determine
the degree of the source term interpolant Πf(U) on any time interval In we have
two cases:
(1) For the first µth time intervals In, n = 1, ..., µ the degree of the source term
interpolant Πf(U) is the index of the time interval In minus one i.e. µ = n − 1

except for the first interval I1 where, in order to obtain a linear algorithm, we
need to use a constant interpolant for the predicted values and then use it in the
linear interpolant for the corrected values. Then the interpolants on the first time
interval I1 are constant and linear µ = 0 and µ = 1 respectively, and on the second
time interval I2 is linear µ = 1 and so on until the µth time interval Iµ where the
interpolant degree will be µ − 1.
(2) The interpolant degree Πf(U) on all the remaining intervals Iµ+1, Iµ+2, ..., IN
is µ.
(I) The solution on the first time interval I1 when n = 1.
To proceed with solution process we start from the first time interval I1. On
this interval we need to construct the linear interpolant Π1
1f(U) = f(U−
0 )ξ0(t) +
f(U−
1 )ξ1(t). We will face the problem that we do not have the solution values
vector U−
1 , hence using this would result into a nonlinear system. To overcome
this difficulty we will use the prediction–correction procedure to attain the required
correct accuracy. We define the time polynomial solution function ¯U|I1 ∈ X1 of
order 1 such that ¯U = 1
j=0 ξj(t)f( ¯U−
j ) and ¯U−
0 = u0.
Now, we need to solve the following problem to obtain the value ¯U−
1 : Indeed,
I1
( ¯U , v)H + a( ¯U, v) dt + ([ ¯U]0, v+
0 )H =
I1
(Π0
0f( ¯U), v)H dt, ∀v ∈ V1, (A.8)
here we approximate f( ¯U) by the constant interpolant Π0
0f( ¯U) = f(., 0, u0) i.e.
µ = 0, which implies that
I1
( ¯U , v)H + a( ¯U, v) dt + ([ ¯U]0, v+
0 )H =
I1
(f( ¯U−
0 ), v)H dt, ∀v ∈ V1. (A.9)

In matrix form, this predictive step yields the following linear system:










0,0M + k1β0,0S 0,1M + k1β0,1S · · · 0,rM + k1β0,rS
1,0M + k1β1,0S 1,1M + k1β1,1S · · · 1,rM + k1β1,rS
...
...
...
...




















¯U
0
1
¯U
1
1
...
¯U
r
1










=










σ0MU
(0)
0
σ1MU
(0)
0
...
σrMU
(0)
0










+










0 k1 0,0M
0 k1 1,0M
...
0 k1 r,0M













0
f(U−
0 )


 . (A.10)
Actually, we just need the predictive value of ¯U−
1 to use it in the next step to solve
for the value U−
1 , the value of ¯U−
0 will not be used. We then use these predictive
solution values to solve the following problem for the corrected solutions values
U0
1 and U1
1 i.e. solving for U|I1 ∈ X1 such that U−
0 = u0:
I1
((U , ν)H + a(U, )) dt + ([U]0, v+
0 )H =
I1
(Π1
1f( ¯U), v)H dt, ∀v ∈ V1. (A.11)
Here, we also choose the interpolant as a linear polynomial Π1
1f( ¯U) = f( ¯U
−
0 )ξ0(t)+
f( ¯U
−
1 )ξ1(t) and now the equation (A.11) implies to the following linear system










0,0M + k1β0,0S 0,1M + k1β0,1S · · · 0,rM + k1β0,rS
1,0M + k1β1,0S 1,1M + k1β1,1S · · · 1,rM + k1β1,rS
...
...
...
...




















U0
1
U1
1
...
Ur
1










=










σ0MU
(0)
0
σ1MU
(0)
0
...
σrMU
(0)
0










+










k1 0,0M k1 0,1M
k1 1,0M k1 1,1M
...
k1 r,0M k1 r,1M













f(U−
0 )
f( ¯U
−
1 )


 . (A.12)
(II) The solution on the second time interval I2 when n = 2 .

We proceed now to the second interval I2 and the interpolant is also taken as a
linear polynomial
(Πf(U), v)H = (Π1
1f(U), v)H on I2,
where
Π1
1f(U) = f(U−
0 )ξ0(t) + f(U−
1 )ξ1(t),
by solving the following problem
I2
((U , v)H + a(U, v)) dt + ([U]1, v+
1 )H =
I2
(Π1
1f(U), v)H dt, ∀v ∈ V2, (A.13)
we get the following linear system










0,0M + k2β0,0S 0,1M + k2β0,1S · · · 0,rM + k2β0,rS
1,0M + k2β1,0S 1,1M + k2β1,1S · · · 1,rM + k2β1,rS
...
...
...
...




















U0
2
U1
2
...
Ur
2










=










σ0MU
(0)
1
σ1MU
(0)
1
...
σrMU
(0)
1










+










k2 0,0M k2 0,1M
k2 1,0M k2 1,1M
...
k2 r,0M k2 r,1M













f(U−
0 )
f(U−
1 )


 . (A.14)
(III) The solution on the (µ + 1)th time interval Iµ+1 when n = µ + 1.
Now, we can solve on the time interval Iµ+1 by using the µth degree interpolant
(Πf(U), v)H = (Πµ
µf(U), v)H on Iµ+1,
where
Πµ
µf(U) = f(U−
0 )ξ0(t) + f(U−
1 )ξ1(t) + ... + f(U−
µ )ξµ(t),
we solve now the following problem on the interval Iµ+1
Iµ+1
((U , v)H + a(U, v)) dt + ([U]µ, v+
µ )H =
Iµ+1
(Πµ
µf(U), v)H dt, ∀v ∈ Vµ+1,
(A.15)

which results to the linear system given in (A.1) for  = 1.
We now conclude with a few relevant examples of the general scheme detailed
above.
Example 1: dG(0) with two–point Gauss–Lobatto quadrature rule (dG(0)-
QGL(2)).
The two–point Gauss–Lobatto quadrature rule on the reference interval ˆI = [0, 1]
is:
QGL(2) =



ˆt0 = 0, ˆt1 = 1,
ˆw0 = 1
2
, ˆw1 = 1
2
.
When r = 0, we have
(U−
n , v)H + kna(U−
n , v) = (U−
n−1, v)H + kn(f(., tn−, U−
n−), v)H,
∀v ∈ V, t ∈ (0, T], (A.16)
which implies that
(M + knS)U−
n = MU−
n−1 + knFn−, t ∈ (0, T]. (A.17)
When  = 0 we have
1
kn
(U−
n − U−
n−1, v)H + a(U−
n , v) = (f(., tn, U−
n ), v)H, ∀v ∈ X0
n, t ∈ (0, T], (A.18)
which is equivalent to the backward (implicit) Euler method and here we need
to solve the nonlinear term by using Newton method or by any other suitable
method.
When  = 1 then we have
1
kn
(U−
n − U−
n−1, v)H + a(U−
n , v) = (f(., tn−1, U−
n−1), v)H, ∀v ∈ X0
n, t ∈ (0, T],(A.19)

which is equivalent to the forward (explicit) Euler method which can be solved
directly.
Example 2: dG(1) with three-point Gauss–Lobatto quadrature rule
(dG(1)-QGL(3)) We will give below some details about the basis and reference
functions.
The Lagrange basis functions corresponding the the time points tn−3, tn−2, tn−1
are
ξn−1(t) = t2−(tn−3+tn−2)t+tn−3tn−2
kn−1(kn−2+kn−1)
, ξn−1(t) = 2t−(tn−3+tn−2)
,
ξn−2(t) = t2−(tn−3+tn−1)t+tn−3tn−1
kn−2kn−1
, ξn−2(t) = 2t−(tn−3+tn−1)
kn−2kn−1
,
ξn−3(t) = t2−(tn−2+tn−1)t+tn−2tn−1
, ξn−3(t) = 2t−(tn−2+tn−1)
.
The mapped functions to the reference interval ˆI = [0, 1] are
ˆξ0(ˆt) = k2
n
ˆt2−(2kn−1+kn−2)knˆt+kn−1(kn−1+kn−2)
, ξ0(ˆt) = 2k2
n
ˆt−(2kn−1+kn−2)kn
,
ˆξ1(ˆt) = −(k2
n
ˆt2−(kn−1+kn−2)knˆt)
kn−2kn−1
, ξ1(ˆt) = −(2k2
n
ˆt−(kn−1+kn−2)kn)
kn−2kn−1
,
ˆξ2(ˆt) = k2
n
ˆt2+kn−1knˆt
, ξ2(ˆt) = 2k2
n
ˆt+kn−1kn
.
In the case of linear function i.e. when r = 1, we have
QGL(3) =



ˆt0 = 0, ˆt1 = 1
2
, ˆt2 = 1,
ˆw0 = 1
6
, ˆw1 = 4
6
, ˆw2 = 1
6
.
Therefore, the reference trial and test functions are linear polynomials in ˆt as
follows:
ˆφ0(ˆt) = (1 − ˆt), ˆφ0(ˆt) = −1,
ˆφ1(ˆt) = ˆt, ˆφ1(ˆt) = 1,

and by following the same steps mentioned in the previous section, we end with
the following linear system



1
2
M + kn
3
S 1
2
M + kn
6
S
−1
2
M + kn
6
S 1
2
M + kn
3
S






U0
n
U1
n


 =



MU0
n−1
0



+



1,n−3M 1,n−2M 1,n−1M
2,n−3M 2,n−2M 2,n−1M










f(U0
n−3)
f(U0
n−2)
f(U0
n−1)







, n ≥ 3. (A.20)
(a) The nonlinear implicit case.
When i = 0 the quadratic interpolant can not be used on the ﬁrst interval I1.
On I1 we will use a linear interpolant while on the other intervals we will proceed
with the quadratic interpolant.
(1) The solution process on the ﬁrst time interval I1.
We proceed as described above, and we arrive at the following linear system



0,0M + k1β0,0S 0,1M + k1β0,1S
1,0M + k1β1,0S 1,1M + k1β1,1S






¯U
0
1
¯U
1
1


 =



σ0MU−
0
σ1MU−
0


 +



0 k1ξ0M
0 k1ξ1M






0
f(U−
0 )


 . (A.21)
By solving this linear system for ¯U
1
1 ( ¯U
0
1 is known), we obtain



1
2
M + k1
3
S 1
2
M + k1
6
S
−1
2
M + k1
6
S 1
2
M + k1
3
S






¯U
0
1
¯U
1
1


 =



MU−
0
0


 +



0 k1
2
M
0 k1
2
M






0
f(U−
0 )


 . (A.22)

Now, we can solve for U0
1 and U1
1 on I1 to have



1
2
M + k1
3
S 1
2
M + k1
6
S
−1
2
M + k1
6
S 1
2
M + k1
3
S






U0
1
U1
1


 =



MU−
0
0



+



k1
3
M k1
6
M
k1
6
M k1
3
M






f(U−
0 )
f( ¯U
1
1)


 . (A.23)
(b) The semi-implicit case.
When i = 1 the quadratic interpolant can not be used on the ﬁrst two intervals
I1 and I2 respectively. So we need to construct special interpolants for these
intervals.
(1) The solution process on the ﬁrst time interval I1.
Continuing as explained in the previous sections, we end with the required linear
system for U0
1 and U1
1,



1
2
M + k1
3
S 1
2
M + k1
6
S
−1
2
M + k1
6
S 1
2
M + k1
3
S






U0
1
U1
1


 =



MU−
0
0



+



k1
3
M k1
6
M
k1
6
M k1
3
M






f(U−
0 )
f( ¯U
1
1)


 . (A.24)
(2) The solution process on the second time interval I2.
Now, after getting the required nodal solution values on the time interval I1, then
we can construct our linear interpolant in the second time interval I2, to obtain
the following linear system



1
2
M + k2
3
S 1
2
M + k2
6
S
−1
2
M + k2
6
S 1
2
M + k2
3
S






U0
2
U1
2


 =



MU1
1
0



+



−
k2
2
6k1
M k2
k1
(k2
6
+ k1
2
)M
−
k2
2
3k1
M k2
k1
(k2
3
+ k1
2
)M






f(U−
0 )
f(U1
1)


 . (A.25)

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