Analysis of hdg methods for stokes flow

Bernardo Cockburn, Jayadeep Gopalakrishnan, Ngoc Cuong Nguyen, Jaume Peraire, Francisco Javier Sayas

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109 Scopus citations

Abstract

In this paper, we analyze a hybridizable discontinuous Galerkin method for numerically solving the Stokes equations. The method uses polynomials of degree k for all the components of the approximate solution of the gradient-velocity-pressure formulation. The novelty of the analysis is the use of a new projection tailored to the very structure of the numerical traces of the method. It renders the analysis of the projection of the errors very concise and allows us to see that the projection of the error in the velocity superconverges. As a consequence, we prove that the approximations of the velocity gradient, the velocity and the pressure converge with the optimal order of convergence of k+1 in L2 for any k ≥ 0. Moreover, taking advantage of the superconvergence properties of the velocity, we introduce a new element-by-element postprocessing to obtain a new velocity approximation which is exactly divergence-free, H(div)-conforming, and converges with order k + 2 for k ≥ 1 and with order 1 for k = 0. Numerical experiments are presented which validate the theoretical results.

Original languageEnglish (US)
Pages (from-to)723-760
Number of pages38
JournalMathematics of Computation
Volume80
Issue number274
DOIs
StatePublished - 2011

Keywords

  • Discontinuous galerkin methods
  • Hybridized methods
  • Lagrange multipliers
  • Mixed methods
  • Stokes flow

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