Discontinuous Galerkin methods on hp-anisotropic meshes II: a posteriori error analysis and adaptivity

Emmanuil H Georgoulis,Edward Hall,Paul Houston

doi:10.1016/j.apnum.2008.12.008

Abstract

We consider the a posteriori error analysis and hp-adaptation strategies for hp-version interior penalty discontinuous Galerkin methods for second-order partial differential equations with nonnegative characteristic form on anisotropically refined computational meshes with anisotropically enriched elemental polynomial degrees. In particular, we exploit duality based hp-error estimates for linear target functionals of the solution and design and implement the corresponding adaptive algorithms 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 and isotropic and anisotropic polynomial degree enrichment. The superiority of the proposed algorithm in comparison with standard hp-isotropic mesh refinement algorithms and an h-anisotropic/ p-isotropic adaptive procedure is illustrated by a series of numerical experiments.

Highlights

The mathematical modeling of advection, diffusion, and reaction processes arises in many application areas
In this paper and the companion article [8] we develop the error analysis for interior penalty Discontinuous Galerkin finite element methods (DGFEMs) applied to second–order partial differential equations with nonnegative characteristic form on general finite element spaces which are possibly anisotropic in both the local meshsize and the local polynomial degree
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

Summary

Introduction

The mathematical modeling of advection, diffusion, and reaction processes arises in many application areas. In this paper and the companion article [8] we develop the error analysis for interior penalty DGFEMs applied to second–order partial differential equations with nonnegative characteristic form on general finite element spaces which are possibly anisotropic in both the local meshsize and the local polynomial degree. Where f ∈ L2(Ω) and c ∈ L∞(Ω) are real–valued, b = {bi}di=1 is a vector function Lipschitz continuous real–valued whose entries bi functions on Ω , are and a = {aij }di,j=1 is a symmetric matrix whose entries aij are bounded, piecewise continuous real–valued functions defined on Ω , with ζ a(x)ζ ≥ 0 ∀ζ ∈ Rd , a.e. x ∈ Ω Under this hypothesis, (1) is termed a partial differential equation with nonnegative characteristic form. For the well-posedness theory (for weak solutions) of the boundary value problem (1), (3), in the case of homogeneous boundary conditions, we refer to [12, 14]

Meshes and finite element spaces

Interior penalty discontinuous Galerkin method

L2-Orthogonal Projection

A priori error analysis

Concluding remarks