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Incompressible Finite Elements via Hybridization. Part I: The Stokes System in Two Space Dimensions
Mathematics and Statistics Faculty Publications and Presentations
  • Bernardo Cockburn, University of Minnesota - Twin Cities
  • Jay Gopalakrishnan, Portland State University
Document Type
Post-Print
Publication Date
1-1-2005
Subjects
  • Finite element method,
  • Vortex-motion,
  • Boundary value problems,
  • Polynomials
Abstract
In this paper, we introduce a new and efficient way to compute exactly divergence-free velocity approximations for the Stokes equations in two space dimensions. We begin by considering a mixed method that provides an exactly divergence-free approximation of the velocity and a continuous approximation of the vorticity. We then rewrite this method solely in terms of the tangential fluid velocity and the pressure on mesh edges by means of a new hybridization technique. This novel formulation bypasses the difficult task of constructing an exactly divergence-free basis for velocity approximations. Moreover, the discrete system resulting from our method has fewer degrees of freedom than the original mixed method since the pressure and the tangential velocity variables are defined just on the mesh edges. Once these variables are computed, the velocity approximation satisfying the incompressibility condition exactly, as well as the continuous numerical approximation of the vorticity, can at once be obtained locally. Moreover, a discontinuous numerical approximation of the pressure within elements can also be obtained locally. We show how to compute the matrix system for our tangential velocity-pressure formulation on general meshes and present in full detail such computations for the lowest-order case of our method.
Description

This is the author’s version of a work that was accepted for publication in SIAM Journal on Numerical Analysis. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. A definitive version was subsequently published in SIAM Journal on Numerical Analysis, 43(4), 1627–1650, and can be found online at: http://epubs.siam.org/doi/pdf/10.1137/04061060X

DOI
10.1137/04061060X
Persistent Identifier
http://archives.pdx.edu/ds/psu/10914
Citation Information
Bernardo Cockburn and Jay Gopalakrishnan. "Incompressible Finite Elements via Hybridization. Part I: The Stokes System in Two Space Dimensions" (2005)
Available at: http://works.bepress.com/jay-gopalakrishnan/80/