hp-Version Discontinuous Galerkin Methods on Polygonal and Polyhedral Meshes [Paperback]

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.  

1 Introduction.- 2 Introduction to Discontinuous Galerkin Methods.- 3 hpInverse and Approximation Estimates.- 4 DGFEMs for Pure Diffusion Problems.- 5 DGFEMs for secondorder PDEs of mixedtype.- 6 Implementation Aspects.- 7 Adaptive Mesh Refinement.- 8 Summary and Outlook.- References.
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,l³’