Ebook: Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications

This book discusses a family of computational methods, known as discontinuous Galerkin methods, for solving partial differential equations. While these methods have been known since the early 1970s, they have experienced an almost explosive growth interest during the last ten to fifteen years, leading both to substantial theoretical developments and the application of these methods to a broad range of problems.

These methods are different in nature from standard methods such as finite element or finite difference methods, often presenting a challenge in the transition from theoretical developments to actual implementations and applications.

This book is aimed at graduate level classes in applied and computational mathematics. The combination of an in depth discussion of the fundamental properties of the discontinuous Galerkin computational methods with the availability of extensive software allows students to gain first hand experience from the beginning without eliminating theoretical insight.

Jan S. Hesthaven is a professor of Applied Mathematics at Brown University.

Tim Warburton is an assistant professor of Applied and Computational Mathematics at Rice University.

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The text offers an introduction to the key ideas, basic analysis, and efficient implementation of discontinuous Galerkin finite element methods (DG-FEM) for the solution of partial differential equations. All key theoretical results are either derived or discussed, including an overview of relevant results from approximation theory, convergence theory for numerical PDE's, orthogonal polynomials etc. Through embedded Matlab codes, the algorithms are discussed and implemented for a number of classic systems of PDE's, e.g., Maxwell's equations, Euler equations, incompressible Navier-Stokes equations, and Poisson- and Helmholtz equations. These developments are done in detail in one and two dimensions on general unstructured grids with high-order elements and all essential routines for 3D extensions are also included and discussed briefly. The three appendices contain an overview of orthogonal polynomials and associated library routines used throughout, a brief introduction to grid generation, and an overview of the associated software (where to get it, list of variables etc).