 
Summary: arXiv:1109.3668v1[math.NA]16Sep2011
MIXED FINITE ELEMENT APPROXIMATION OF THE VECTOR
LAPLACIAN WITH DIRICHLET BOUNDARY CONDITIONS
DOUGLAS N. ARNOLD, RICHARD S. FALK, AND JAY GOPALAKRISHNAN
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.
1. Introduction
We consider the vector Laplace equation (Hodge Laplace equation for 1forms) on a two
dimensional domain . That is, given a vector field f on , we seek a vector field u such
that
(1.1) curl rot u  grad div u = f in .
