Title: Mixed finite element approximation of the vector Laplacian with
Dirichlet boundary conditions
Authors: Douglas N. Arnold, Richard S. Falk, and Jay Gopalakrishnan
Source: Math. Models and Methods in Applied Sciences, 22 (2012),
No. 9: 26 pages.
Status: Published
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.