 
Summary: 1
Preconditioning in H(div) and
Applications
Douglas N. Arnold 1
, Richard S. Falk 2
and Ragnar
Winther3 4
Abstract. Summarizing the work of [AFW97], we show how to construct
preconditioners using domain decomposition and multigrid techniques for 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. These
preconditioners are shown to be spectrally equivalent to the inverse of the operator and
thus may be used to precondition iterative methods so that any given error reduction
may be achieved in a finite number of iterations 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.
1.1 Introduction
This paper summarizes the work of [AFW97], in which we consider the solution of the
system of linear algebraic equations which arises from the finite element discretization
of boundary value problems in two space dimensions for the differential operator
