Hybridized globally divergence-free LDG methods. Part I: The stokes problem

Jesús Carrero, Bernardo Cockburn, Dominik Schötzau

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76 Scopus citations

Abstract

We devise and analyze a new local discontinuous Galerkin (LDG) method for the Stokes equations of incompressible fluid flow. This optimally convergent method is obtained by using an LDG method to discretize a vorticity-velocity formulation of the Stokes equations and by applying a new hybridization to the resulting discretization. One of the main features of the hybridized method is that it provides a globally divergence-free approximate velocity without having to construct globally divergence-free finite-dimensional spaces; only elementwise divergence-free basis functions are used. Another important feature is that it has significantly less degrees of freedom than all other LDG methods in the current literature; in particular, the approximation to the pressure is only defined on the faces of the elements. On the other hand, we show that, as expected, the condition number of the Schur-complement matrix for this approximate pressure is of order h -2 in the mesh size h. Finally, we present numerical experiments that confirm the sharpness of our theoretical a priori error estimates.

Original languageEnglish (US)
Pages (from-to)533-563
Number of pages31
JournalMathematics of Computation
Volume75
Issue number254
DOIs
StatePublished - Apr 2006

Keywords

  • Divergence-free elements
  • Hybridized methods
  • Local discontinuous Galerkin methods
  • Stokes equations

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