A note on the design of hp-version interior penalty discontinuous Galerkin finite element methods for degenerate problems

Emmanuil H Georgoulis,Andris Lasis

doi:10.1093/imanum/dri038

Abstract

We consider a variant of the hp-version interior penalty discontinuous Galerkin finite element method (IP-DGFEM) for second-order problems of degenerate type. We do not assume uniform ellipticity of the diffusion tensor. Moreover, diffusion tensors of arbitrary form are covered in the theory presented. A new, refined recipe for the choice of the discontinuity-penalization parameter (that is present in the formulation of the IP-DGFEM) is given. Making use of the recently introduced augmented Sobolev space framework, we prove an hp-optimal error bound in the energy norm and an h-optimal and slightly p-suboptimal (by only half an order of p) bound in the L 2 norm (the latter, for the symmetric version of the IP-DGFEM), provided that the solution belongs to an augmented Sobolev space.

Highlights

The numerical solution of boundary value problems containing second order equations of degenerate type is of interest, as in many models diffusion can be small, degenerate, or even identically equal to zero in subregions of the domain under consideration
Strang’s Second Lemma is employed for the error analysis. We show that this modification delivers an hp-optimal error bound in the energy norm and and h-optimal and slightly p-suboptimal bound in the L2 norm when the solution admits sufficient regularity in the context of augmented Sobolev spaces, presented in [7]
The error analysis of the hp-version IP-DGFEM is discussed in Section 4: we prove an hp-optimal a priori error bound in the energy norm and an h-optimal and slightly p-suboptimal bound in the L2 norm, using the projection error bounds introduced in [7]

Summary

Introduction

The numerical solution of boundary value problems containing second order equations of degenerate type is of interest, as in many models diffusion can be small, degenerate, or even identically equal to zero in subregions of the domain under consideration. We show that this modification delivers an hp-optimal error bound in the energy norm and and h-optimal and slightly p-suboptimal (by only half an order of p) bound in the L2 norm when the solution admits sufficient regularity in the context of augmented Sobolev spaces, presented in [7]. The present analysis is inspired by the analysis by Perugia and Schotzau [11] in the case of the local DG method for linear elliptic problems, where an inconsistent weak formulation of the problem is derived (through the use of lifting operators) and Strang’s Second Lemma is employed for the error analysis.

Model problem and preliminaries

Discontinuous Galerkin finite element methods

Error analysis

Concluding remarks