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Analysis of HDG Methods for Stokes Flow
Mathematics of Computation
  • Bernardo Cockburn, University of Minnesota - Twin Cities
  • Jay Gopalakrishnan, Portland State University
  • Ngoc Cuong Nguyen, Massachusetts Institute of Technology
  • Jaume Peraire, Massachusetts Institute of Technology
  • Francisco-Javier Sayas, Departamento de Matemática Aplicada
Document Type
Publication Date
  • Galerkin methods,
  • Stokes flow,
  • Discontinuous functions,
  • Numerical analysis,
  • Approximation theory,
  • Polynomials
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 $ L[superscript 2]$ for any $ k [greater than or equal to] 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, $ \mathbf{H}($div$ )$-conforming, and converges with order $ k+2$ for $ k[greater than or equal to]1$ and with order $ 1$ for $ k=0$. Numerical experiments are presented which validate the theoretical results.

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© 2011 American Mathematical Society.

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Citation Information
Cockburn, Bernardo et al. “Analysis of HDG methods for Stokes flow.” Mathematics of Computation 80.274 (2011): 723-723.