Boundary-conforming discontinuous galerkin methods via extensions from subdomains

Bernardo Cockburn, Deepa Gupta, Fernando Reitich

Research output: Contribution to journalArticlepeer-review

12 Scopus citations

Abstract

A new way of devising numerical methods is introduced whose distinctive feature is the computation of a finite element approximation only in a polyhedral subdomain D of the original, possibly curved-boundary domain. The technique is applied to a discontinuous Galerkin method for the one-dimensional diffusion-reaction problem. Sharp a priori error estimates are obtained which identify conditions, on the subdomain D and the discretization parameters of the discontinuous Galerkin method, under which the method maintains its original optimal convergence properties. The error analysis is new even in the case in which D=Ω. It allows to see that the uniform error at any given interval is bounded by an interpolation error associated to the interval plus a significantly smaller error of a global nature. Numerical results confirming the sharpness of the theoretical results are displayed. Also, preliminary numerical results illustrating the application of the method to two-dimensional second-order elliptic problems are shown.

Original languageEnglish (US)
Pages (from-to)144-184
Number of pages41
JournalJournal of Scientific Computing
Volume42
Issue number1
DOIs
StatePublished - Jan 2010

Bibliographical note

Funding Information:
B. Cockburn was supported in part by the National Science Foundation (Grant DMS-0712955) and by the University of Minnesota Supercomputing Institute.

Keywords

  • Discontinuous Galerkin methods
  • Elliptic problems

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