  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.
