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.