We consider the finite element solution of the vector Laplace equation on a
domain in two dimensions.  For some choices of boundary conditions, there is a
theory, making use of finite element differential complexes and bounded
cochain projections, that shows that a mixed finite element method using
appropriate choices of finite element spaces, and in which the rotation
of the solution is introduced as a second unknown, leads 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 present
computational examples that demonstrate that such a mixed finite element method
does not perform optimally in this case, and an analysis which theoretically
confirms the suboptimal convergence that does occur and indicates the source
of the problem.  These results also have implications for the solution of the
biharmonic equation and of the Stokes equations using a mixed formulation
involving the vorticity.