Discontinuous Galerkin Methods for Advection-Diffusion-Reaction Problems on Anisotropically Refined Meshes

Emmanuil H Georgoulis,Edward Hall,Paul Houston

doi:10.1137/060672352

Abstract

In this paper we consider the a posteriori and a priori error analysis of discontinuous Galerkin interior penalty methods for second-order partial differential equations with nonnegative characteristic form on anisotropically refined computational meshes. In particular, we discuss the question of error estimation for linear target functionals, such as the outflow flux and the local average of the solution. Based on our a posteriori error bound, we design and implement the corresponding adaptive algorithm to ensure reliable and efficient control of the error in the prescribed functional to within a given tolerance. This involves exploiting both local isotropic and anisotropic mesh refinement. The theoretical results are illustrated by a series of numerical experiments.

Highlights

The mathematical modeling of advection, diffusion, and reaction processes arises in many application areas
The a priori error estimation is based on exploiting the analysis developed in [13], which assumed that the underlying computational mesh is shape–regular, together with an extension of the techniques developed in [10] which precisely describe the anisotropy of the mesh; for related anisotropic approximation results, we refer to [1, 22, 21, 6], for example
New, sharp directionally-sensitive bounds have been derived for the polynomial approximation on anisotropic elements exploiting the ideas presented in [10], and subsequently generalizing the results of that paper. These new anisotropic polynomial approximation results have been exploited in the proceeding a priori analysis of the numerical error for general linear target functionals of the solution on anisotropic meshes

Summary

Introduction

The mathematical modeling of advection, diffusion, and reaction processes arises in many application areas. For all κ in Th uniformly throughout the mesh for some positive constants C1, C2, and C3 This will be important as our error estimates will be expressed in terms of Sobolev norms over the element domains κ, in order to ensure that only the scaling and orientation introduced by the affine element maps Fκ are present in the analysis.

Cinv mf mκ v

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Findings

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