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2017 | Buch

Introduction Kapitel lesen Erstes Kapitel lesen

hp-Version Discontinuous Galerkin Methods on Polygonal and Polyhedral Meshes

verfasst von: Dr. Andrea Cangiani, Dr. Zhaonan Dong, Dr. Emmanuil H. Georgoulis, Prof. Paul Houston

Verlag: Springer International Publishing

Buchreihe : SpringerBriefs in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Introduction
Abstract
In this chapter we present a historical overview of discontinuous Galerkin finite element methods; in particular, we discuss their key properties and relative advantages compared to other schemes employed for the numerical approximation of partial differential equations. In addition, we highlight the key computational advantages of exploiting general computational meshes consisting of polygonal/ polyhedral elements, in terms of both meshing complicated geometries in an affordable manner, as well as providing sequences of coarse geometry-conforming meshes needed for the design of efficient multi-level solvers. Finally, we outline some standard notation used throughout this volume.
Andrea Cangiani, Zhaonan Dong, Emmanuil H. Georgoulis, Paul Houston
Chapter 2. Introduction to Discontinuous Galerkin Methods
Abstract
The purpose of this chapter is to present an overview of the construction of discontinuous Galerkin finite element methods for a general class of second-order partial differential equations with nonnegative characteristic form. This class of equations includes second-order elliptic and parabolic partial differential equations, ultra-parabolic equations, first-order hyperbolic problems, the Kolmogorov-Fokker-Planck equations of Brownian motion, the equations of boundary layer theory in hydrodynamics, and various other degenerate equations.
Andrea Cangiani, Zhaonan Dong, Emmanuil H. Georgoulis, Paul Houston
Chapter 3. Inverse Estimates and Polynomial Approximation on Polytopic Meshes
Abstract
This chapter develops the key mathematical tools needed to study the stability and convergence properties of hp-version discontinuous Galerkin finite element methods on polytopic meshes. A key issue in this setting is that general shape-regular polytopic meshes in \(\mathbb{R}^{d}\), d > 1, may, under mesh refinement, possess elements with (d − k)-dimensional facets, k = 1, 2, …, d − 1, which degenerate as the mesh size tends to zero. Thereby, care must be taken to ensure that the resulting inverse estimates and polynomial approximation results are sensitive to this type of degeneracy. The key approach adopted here is to exploit known results for standard elements, both within an L 2- and L ∞ -setting, and to take the minimum of the resulting bounds. In this way, bounds which are optimal in both the h-version and p-version setting may be deduced, which directly account for (d − k)-dimensional facet degeneration, k = 1, 2, …, d − 1.
Andrea Cangiani, Zhaonan Dong, Emmanuil H. Georgoulis, Paul Houston
Chapter 4. DGFEMs for Pure Diffusion Problems
Abstract
In this chapter we study the stability and hp-version a priori error analysis of the discontinuous Galerkin finite element discretization of a pure diffusion problem. In particular, we develop the underlying theory for two different sets of shape assumptions which the polytopic elements forming the computational mesh must satisfy. In the first instance, we assume that the number of faces each element possesses remains uniformly bounded under mesh refinement, but without a restriction concerning shape-regularity. Secondly, we pursue the analysis in the case when this assumption is violated, i.e., when polytopic elements are permitted to have an arbitrary number of faces under mesh refinement; however, in this setting, a generalized shape-regularity assumption must be satisfied. The relationship between these different mesh assumptions is discussed in detail; indeed, the combination of these conditions allows for very general polytopic meshes to be admitted within our analysis.
Andrea Cangiani, Zhaonan Dong, Emmanuil H. Georgoulis, Paul Houston
Chapter 5. DGFEMs for Second-Order PDEs of Mixed-Type
Abstract
This chapter is devoted to studying the stability and convergence properties of the discontinuous Galerkin finite element method (DGFEM) for the numerical approximation of a general class of second-order partial differential equations (PDEs) with nonnegative characteristic form. While this class of second-order equations naturally includes parabolic equations, we also pursue the analysis of parabolic time-dependent PDEs in a separate manner in order to admit the use of local time-stepping algorithms. The general analysis of DGFEMs for PDEs with nonnegative characteristic form is pursued under the assumption that the number of faces each element possesses remains bounded under mesh refinement. For the special subclass of parabolic problems we adopt the assumption which permits an arbitrary number of faces per element, thereby highlighting that both assumptions lead to rigorous a priori error estimates for DGFEMs applied to parabolic problems.
Andrea Cangiani, Zhaonan Dong, Emmanuil H. Georgoulis, Paul Houston
Chapter 6. Implementation Aspects
Abstract
In this chapter we outline the key implementation aspects of exploiting discontinuous Galerkin finite element methods on general computational meshes consisting of polytopic elements: mesh generation, construction of the elemental polynomial basis, and numerical integration. We also present some numerical examples to highlight the sharpness of the a priori error bounds derived in this volume for both a steady advection-diffusion-reaction problem and a (degenerate) parabolic problem.
Andrea Cangiani, Zhaonan Dong, Emmanuil H. Georgoulis, Paul Houston
Chapter 7. Adaptive Mesh Refinement
Abstract
This chapter is devoted to the automatic adaptive mesh refinement of polytopic meshes generated based on exploiting agglomeration of a given background geometry-conforming fine mesh. In particular, we exploit mesh partitioning techniques to design the underlying coarse mesh as well as for subdividing agglomerated elements marked for refinement. Here, the underlying adaptive refinement algorithm is constructed by employing goal-oriented dual-weighted-residual a posteriori error estimation techniques; in this setting, the aim of the computation is to accurately approximate the value of a given target or output functional of the solution. Numerical examples are presented to highlight the potential benefits of this approach when approximating partial differential equations posed on complicated domains.
Andrea Cangiani, Zhaonan Dong, Emmanuil H. Georgoulis, Paul Houston
Chapter 8. Summary and Outlook
Abstract
In this volume we have presented an overview of recent developments concerning the mathematical analysis and practical application of DGFEMs on general meshes consisting of polytopic elements with a variable number of faces per element. The use of such general computational meshes in conjunction with DGFEMs has a number of key advantages; most notably: the number of degrees of freedom within the underlying finite element space is independent of the complexity of the geometry. Thereby, coarse approximations, which may possess sufficient accuracy for engineering applications, may be computed in an efficient manner. Moreover, adaptivity may subsequently be employed to improve solution accuracy by adding resolution in regions of the computational domain which directly affect output quantities of interest. The flexibility of DGFEMs means that the underlying discretization scheme naturally admits high-order polynomial degrees. While not directly discussed within this volume, the exploitation of general polytopic meshes, and in particular, meshes employing agglomerated elements, are essential for the design of efficient multi-level solvers; for recent work in this direction, we refer to [9, 13, 15], for example.
Andrea Cangiani, Zhaonan Dong, Emmanuil H. Georgoulis, Paul Houston
Backmatter
Metadaten
Titel
hp-Version Discontinuous Galerkin Methods on Polygonal and Polyhedral Meshes
verfasst von
Dr. Andrea Cangiani
Dr. Zhaonan Dong
Dr. Emmanuil H. Georgoulis
Prof. Paul Houston
Copyright-Jahr
2017
Electronic ISBN
978-3-319-67673-9
Print ISBN
978-3-319-67671-5
DOI
https://doi.org/10.1007/978-3-319-67673-9