Mahboub Baccouch

By Mahboub Baccouch

We present an analysis of the discontinuous Galerkin (DG) finite element method for nonlinear ordinary differential equations (ODEs). We prove that the DG solution is $(p + 1) $th order convergent in the $L^2$-norm, when the space of piecewise polynomials of degree $p$ is used. A $ (2p+1) $th order superconvergence rate of the DG approximation at the downwind point of each element is obtained under quasi-uniform meshes. Moreover, we prove that the DG solution is superconvergent with order $p+2$ to a particular projection of the exact solution. The superconvergence results are used to show that the leading term of the DG error is proportional to the $ (p + 1) $-degree right Radau polynomial. These results allow us to develop a residual-based a posteriori error estimator which is computationally simple, efficient, and asymptotically exact. The proposed a posteriori error estimator is proved to converge to the actual error in the $L^2$-norm with order $p+2$. Computational results indicate that the theoretical orders of convergence are optimal. Finally, a local adaptive mesh refinement procedure that makes use of our local a posteriori error estimate is also presented. Several numerical examples are provided to illustrate the global superconvergence results and the convergence of the proposed estimator under mesh refinement.

Part of the book: Perusal of the Finite Element Method

By Mahboub Baccouch

The finite element (FE) method is a numerical technique for computing approximate solutions to complex mathematical problems described by differential equations. The method was developed in the 1950s to solve complicated problems in engineering, notably in elasticity and structural mechanics modeling involving elliptic partial differential equations and complicated geometries. But nowadays the range of applications is quite extensive. In particular, the FE method has been successfully applied to many problems such as fluid–structure interaction, thermomechanical, thermochemical, thermo-chemo-mechanical problems, biomechanics, biomedical engineering, piezoelectric, ferroelectric, electromagnetics, and many others. This chapter contains a summary of the FE method. Since the remaining chapters of this textbook are based on the FE method, we present it in this chapter as a method for approximating solutions of ordinary differential equations (ODEs) and partial differential equations (PDEs).

Part of the book: Finite Element Methods and Their Applications

By Mahboub Baccouch

The Burgers equation is the simplest equation combining both nonlinear propagation effects and diffusive effects. It describes a weak shock phenomena in gas dynamics. This equation has been a center of interest for researchers studying various physical phenomena such as theory of shock waves, fluid dynamics, turbulent flow and gas dynamics. In this chapter, the viscid and inviscid Burgers equations are studied. Two important mathematical models described by the Burgers equations are presented. Then a very brief review of theoretical results for conservation laws are provided. In particular, the characteristic method is used to demonstrate the development of shocks. Furthermore, the notion of weak solutions, state the Rankine-Hugoniot jump condition, and introduce the notion of an entropy condition are introduced. Finally, several important numerical methods to approximate the solution to the viscid and inviscid Burgers equations are developed. For the inviscid Burgers equation, the classical finite difference and finite volume methods are presented in details including Up-wind nonconservative, Up-wind conservative, Lax-Friedrichs, Lax-Wendroff, MacCormack, Godunov methods. In addition, the discontinuous Galerkin method is described in details. For the viscid Burgers equation, a finite difference method, a Chebyshev collocation method, an ultra-weak discontinuous Galerkin method, and a local discontinuous Galerkin method are presented.

Part of the book: Computational Fluid Dynamics

By Mahboub Baccouch

The Finite Element Method (FEM) is a numerical framework commonly employed to approximate solutions of differential equations that describe complex physical and engineering phenomena. Originating in the 1950s as a tool for addressing difficult problems in structural mechanics and elasticity – particularly those governed by elliptic partial differential equations (PDEs) – it has since evolved into a general numerical framework applicable to a broad range of scientific and engineering disciplines. Today, FEM is employed in diverse areas such as electromagnetics, fluid dynamics, structural analysis, heat transfer, aerospace, biomechanics, automotive, computer graphics, biomedical research, manufacturing, and construction. This chapter offers a concise introduction to the fundamental principles of FEM, serving as the foundation for the topics developed in later chapters. Our goal is to present FEM as a unified approach for approximating the solutions of key classes of PDEs. In particular, we apply the method to second-order elliptic PDEs, the heat (or diffusion) equation, the reaction-convection-diffusion equation, the wave equation, and the Navier-Stokes equations.

Part of the book: Finite Element Method Applications