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

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.

Discontinuous Galerkin methods on hp-anisotropic meshes I: a priori error analysis

“…employing the definition of the discontinuity-penalization parameter ϑ stated in (16), together with the bounded variation conditions, and applying the approximation results from Section 3, the result follows after rearranging the terms involved.2 Remark 4.6 The above result represents an extension of the a priori error bounds derived in the articles [7,10] to the case when general anisotropic computational meshes are employed and anisotropic local polynomial degrees are allowed.…”

“…The aim of this section is to develop the a priori error analysis for general linear target functionals J(·) of the solution; for related work, we refer to [2,7,10], for example. We begin by defining the energy norm | ·| by…”

“…While the companion article [8] will be concerned with the derivation of computable a posteriori error bounds, and their implementation within automatic hp-anisotropic adaptive software, the current paper will focus on a priori error estimation. To this end, the proofs of the a priori error bounds presented in this article are based on exploiting the analysis developed in [10], which assumed that the underlying computational mesh is shape-regular and that the polynomial approximation orders are isotropic, together with the anisotropic hp-approximation results presented in [6]; for related work on anisotropic approximation theory, see [1,3,4,7,15,16], for example, and the references cited therein. We also refer to the recent article [7], where the goal-oriented error analysis of the interior penalty DGFEM on anisotropic computational meshes, assuming that the polynomial degree is kept fixed, has been developed.…”

A Class of Domain Decomposition Preconditioners for hp-Discontinuous Galerkin Finite Element Methods

“…Finally, we assume that the following bounded local variation property holds (cf. [25,26]): for any pair of elements…”

A higher‐order discontinuous enrichment method for the solution of high péclet advection–diffusion problems on unstructured meshes

“…In general, solutions of the form given in (65) are not in the span of the enrichment space V E described in (23), except for certain values of φ and ψ. Here, the advection direction is fixed to φ = π/7 and the direction ψ is varied by angles of π/4 so that the exact solution (65) is not contained in the space of approximation of any of the DGM element considered herein.…”