Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations

Excerpt

This book is an introduction to a family of discontinuous Galerkin (DG) methods applied to some steady-state and time-dependent model problems. A special effort was made to have the material self-contained as much as possible. The book is well suited to numerical analysts interested in DG methods but also to applied mathematicians who study CFD or porous media flow. Practical implementation issues are discussed, which can be of interest to engineers. The material can be used in a graduate level course on the numerical solution of partial differential equations. Chapter 1 is introductory and can be used in a scientific computing class for senior undergraduate students. Prerequisites are calculus and linear algebra.

In this book, we mainly focus on the class of primal DG methods, namely variations of interior penalty methods. In the text, these methods are referred to as the symmetric interior penalty Galerkin (SIPG), incomplete interior penalty Galerkin (IIPG), and nonsymmetric interior penalty Galerkin (NIPG) methods. The book is divided into three parts: Part I focuses on the application of DG to second order elliptic problems in one dimension first and then in higher dimensions. In Part II, the time-dependent parabolic problems (without and with convection) are presented. Finally, Part III covers some applications of DG to solid mechanics (linear elasticity), to fluid dynamics (Stokes and Navier—Stokes), and to porous media flow (two-phase and miscible displacement).

We try to discuss both theoretical and computational aspects of the DG methods. In particular, for the elliptic equations, a code written in MATLAB® for one-dimensional problems is provided in Appendix B.1. The text contains algorithms for the implementation of DG methods in two or three dimensions. Corresponding routines written in C are provided in Appendix B.2 as well.