 
Summary: MATHEMATICS OF COMPUTATION
Volume 66, Number 219, July 1997, Pages 957984
S 00255718(97)008260
PRECONDITIONING IN H (div) AND APPLICATIONS
DOUGLAS N. ARNOLD, RICHARD S. FALK, AND R. WINTHER
Dedicated to Professor Ivo Babuska on the occasion of his seventieth birthday.
Abstract. We consider the solution of the system of linear algebraic equa
tions 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 con
struct 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.
1. Introduction
