Hybridized globally divergence-free LDG methods. Part I: The Stokes problem
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- by Jesús Carrero, Bernardo Cockburn and Dominik Schötzau PDF
- Math. Comp. 75 (2006), 533-563 Request permission
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.References
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Additional Information
- Jesús Carrero
- Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
- Email: carrero@math.umn.edu
- Bernardo Cockburn
- Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
- Email: cockburn@math.umn.edu
- Dominik Schötzau
- Affiliation: Department of Mathematics, University of British Columbia, Vancouver, British Columbia V6T 1Z2, Canada
- Email: schoetzau@math.ubc.ca
- 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.
- © Copyright 2005 American Mathematical Society
- Journal: Math. Comp. 75 (2006), 533-563
- MSC (2000): Primary 65N30
- DOI: https://doi.org/10.1090/S0025-5718-05-01804-1
- MathSciNet review: 2196980