Title: Preconditioning in H(div) and applications

Authors: Douglas N. Arnold, Richard S. Falk, and R. Winther.

Source: Math. Comp., 66(219):957-984, 1997.

Abstract: We consider the solution of the system of linear algebraic
equations which arises from the finite element discretization of
boundary value problems associated to the differential operator I -
grad div. The natural setting for such problems is in the Hilbert
space H(div) and the variational formulation is based on the inner
product in H(div). We show how to construct preconditioners for these
equations using both domain decomposition and multigrid
techniques. These preconditioners are shown to be spectrally
equivalent to the inverse of the operator. As a consequence, they may
be used to precondition iterative methods so that any given error
reduction may be achieved in a finite number of iterations, with the
number independent of the mesh discretization. We describe
applications of these results to the efficient solution of mixed and
least squares finite element approximations of elliptic boundary value
problems.

Keywords: preconditioner, mixed method, least squares, finite element,
multigrid, domain decomposition

URL: http://www.math.rutgers.edu/~falk/papers/hdiv.pdf