Analysis of variable-degree HDG methods for convection-diffusion equations. Part I: General nonconforming meshes

Yanlai Chen, Bernardo Cockburn

Research output: Contribution to journalArticlepeer-review

35 Scopus citations

Abstract

In this paper the first error analyses of hybridizable discontinuous Galerkin (HDG) methods for convection-diffusion equations for variable-degree approximations and nonconforming meshes are presented. The analysis technique is an extension of the projection-based approach recently used to analyse the HDG method for the purely diffusive case. In particular, for approximations of degree k on all elements and conforming meshes, we show that the order of convergence of the error in the diffusive flux is k + 1 and that of a projection of the error in the scalar unknown is 1 for k = 0 and k + 2 for k > 0. When nonconforming meshes are used our estimates do not rule out a degradation of 1/2 in the order of convergence in the diffusive flux and a loss of 1 in the order of convergence of the projection of the error in the scalar variable. However, they do guarantee the optimal convergence of order k + 1 of the scalar variable.

Original languageEnglish (US)
Pages (from-to)1267-1293
Number of pages27
JournalIMA Journal of Numerical Analysis
Volume32
Issue number4
DOIs
StatePublished - Oct 2012

Bibliographical note

Funding Information:
The second author was partially supported by National Science Foundation (Grants DMS-0712955 and DMS-1115331) and by the University of Minnesota Supercomputing Institute.

Keywords

  • convection-diffusion equations
  • discontinuous Galerkin methods
  • finite element methods
  • hybridization
  • nonconforming meshes
  • superconvergence

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