Superconvergent discontinuous galerkin methods for second-order elliptic problems

Bernardo Cockburn, Johnny Guzmán, Haiying Wang

Research output: Contribution to journalArticlepeer-review

113 Scopus citations

Abstract

We identify discontinuous Galerkin methods for second-order elliptic problems in several space dimensions having superconvergence properties similar to those of the Raviart-Thomas and the Brezzi-Douglas-Marini mixed methods. These methods use polynomials of degree k ≥ 0 for both the potential as well as the flux. We show that the approximate flux converges in L2 with the optimal order of k + 1. and that the approximate potential and its numerical trace superconverge, in L2-like norms, to suitably chosen projections of the potential, with order k + 2. We also apply element-by-element postprocessing of the approximate solution to obtain new approximations of the flux and the potential. The new approximate flux is proven to have normal components continuous across inter-element boundaries, to converge in L 2 with order k + 1, and to have a divergence converging in L 2 also with order k + 1. The new approximate potential is proven to converge with order k + 2 in L2. Numerical experiments validating these theoretical results arc presented.

Original languageEnglish (US)
Pages (from-to)1-24
Number of pages24
JournalMathematics of Computation
Volume78
Issue number265
DOIs
StatePublished - 2009

Keywords

  • Discontinuous galerkin methods
  • Finite element methods
  • Mixed methods
  • Postprocessing
  • Superconvergence

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