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

Authors: Jesús Carrero, Bernardo Cockburn and Dominik Schötzau
Journal: Math. Comp. 75 (2006), 533-563
MSC (2000): Primary 65N30
Published electronically: December 16, 2005
MathSciNet review: 2196980
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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.

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Additional Information

Jesús Carrero
Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455

Bernardo Cockburn
Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455

Dominik Schötzau
Affiliation: Department of Mathematics, University of British Columbia, Vancouver, British Columbia V6T 1Z2, Canada

Keywords: Divergence-free elements, local discontinuous Galerkin methods, hybridized methods, Stokes equations
Received by editor(s): June 9, 2004
Received by editor(s) in revised form: February 9, 2005
Published electronically: December 16, 2005
Additional Notes: The second author was supported in part by the National Science Foundation (Grant DMS-0411254) and by the University of Minnesota Supercomputing Institute. The third author was supported in part by the Natural Sciences and Engineering Research Council of Canada.
Article copyright: © Copyright 2005 American Mathematical Society