hp-Version Discontinuous Galerkin Methods on Polygonal and Polyhedral Meshes

by Andrea Cangiani, Zhaonan Dong, Emmanuil H. Georgoulis, Paul Houston

Overview

Over the last few decades discontinuous Galerkin finite element methods (DGFEMs) have been witnessed tremendous interest as a computational framework for the numerical solution of partial differential equations. Their success is due to their extreme versatility in the design of the underlying meshes and local basis functions, while retaining key features of both (classical) finite element and finite volume methods. Somewhat surprisingly, DGFEMs on general tessellations consisting of polygonal (in 2D) or polyhedral (in 3D) element shapes have received little attention within the literature, despite the potential computational advantages.

This volume introduces the basic principles of hp-version (i.e., locally varying mesh-size and polynomial order) DGFEMs over meshes consisting of polygonal or polyhedral element shapes, presents their error analysis, and includes an extensive collection of numerical experiments. The extreme flexibility provided by the locally variable elemen

t-shapes, element-sizes, and element-orders is shown to deliver substantial computational gains in several practical scenarios.

Product Details

ISBN-13:	9783319676715
Publisher:	Springer International Publishing
Publication date:	11/28/2017
Series:	SpringerBriefs in Mathematics
Edition description:	1st ed. 2017
Pages:	131
Product dimensions:	6.10(w) x 9.25(h) x (d)

Table of Contents

1 Introduction.- 2 Introduction to Discontinuous Galerkin Methods.- 3 hp–Inverse and Approximation Estimates.- 4 DGFEMs for Pure Diffusion Problems.- 5 DGFEMs for second–order PDEs of mixed–type.- 6 Implementation Aspects.- 7 Adaptive Mesh Refinement.- 8 Summary and Outlook.- References.

Editorial Reviews

This book should be very useful to Ph.D students and researchers who are eager to explore and understand the mathematical analysis of DGFEMs on polytopic meshes for second-order elliptic equations. The presentation is lucid and the book provides an extensive list of references in the area.” (Neela Nataraj, Mathematical Reviews, October, 2018)

From the Publisher