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Article
Mixed Finite Element Approximation of the Vector Laplacian with Dirichlet Boundary Conditions
Mathematical Models and Methods in Applied Sciences
  • Douglas N. Arnold, University of Minnesota - Twin Cities
  • Richard S. Falk, Rutgers University
  • Jay Gopalakrishnan, Portland State University
Document Type
Post-Print
Publication Date
1-1-2012
Subjects
  • Finite element method,
  • Approximation theory,
  • Laplacian operator,
  • Biharmonic equations
Disciplines
Abstract
We consider the finite element solution of the vector Laplace equation on a domain in two dimensions. For various choices of boundary conditions, it is known that a mixed finite element method, in which the rotation of the solution is introduced as a second unknown, is advantageous, and appropriate choices of mixed finite element spaces lead to a stable, optimally convergent discretization. However, the theory that leads to these conclusions does not apply to the case of Dirichlet boundary conditions, in which both components of the solution vanish on the boundary. We show, by computational example, that indeed such mixed finite elements do not perform optimally in this case, and we analyze the suboptimal convergence that does occur. As we indicate, these results have implications for the solution of the biharmonic equation and of the Stokes equations using a mixed formulation involving the vorticity.
Description

This is the author’s version of a work that was accepted for publication. The electronic version of this article was published in Mathematical Models and Methods in Applied Sciences, Volume 22, Issue 9. © Copyright World Scientific Publishing Company and is available online at: http://www.worldscientific.com/worldscinet/m3as

DOI
10.1142/S0218202512500248
Persistent Identifier
http://archives.pdx.edu/ds/psu/10606
Citation Information
Douglas N. Arnold, Richard S. Falk and Jay Gopalakrishnan. "Mixed Finite Element Approximation of the Vector Laplacian with Dirichlet Boundary Conditions" Mathematical Models and Methods in Applied Sciences (2012)
Available at: http://works.bepress.com/jay-gopalakrishnan/85/