ESAIM: Mathematical Modelling and Numerical Analysis

Table of Contents - November 2012 - Volume 46, Issue 06  

Research Article

Discontinuous Galerkin methods for problems with Dirac delta source ∗

Paul Houstona1 and Thomas Pascal Wihlera2

a1 School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK. Paul.Houston@nottingham.ac.uk

a2 Mathematics Institute, University of Bern, 3012 Bern, Switzerland; wihler@math.unibe.ch

Abstract

In this article we study discontinuous Galerkin finite element discretizations of linear second-order elliptic partial differential equations with Dirac delta right-hand side. In particular, assuming that the underlying computational mesh is quasi-uniform, we derive an a priori bound on the error measured in terms of the L2-norm. Additionally, we develop residual-based a posteriori error estimators that can be used within an adaptive mesh refinement framework. Numerical examples for the symmetric interior penalty scheme are presented which confirm the theoretical results.

(Received June 17 2011)

(Revised November 22 2011)

(Online publication May 31 2012)

Key Words:

  • Elliptic PDEs;
  • discontinuous Galerkin methods;
  • Dirac delta source

Mathematics Subject Classification:

  • 65N30

Footnotes

∗  Thomas Wihler acknowledges the financial support of the Swiss National Science Foundation (SNF) under Grant No. 200021_126594.