INTERIOR PENALTY PRECONDITIONERS FOR MIXED FINITE 
                 ELEMENT APPROXIMATIONS OF ELLIPTIC PROBLEMS

          TORGEIR RUSTEN, PANAYOT S. VASSILEVSKI, AND RAGNAR WINTHER

                                   Abstract

The purpose of this talk is to discuss the construction of preconditioners for
the discrete problems arising from mixed finite element discretizations of
second order elliptic boundary value problems.  It is necesarry to build a
preconditioner for a nonconforming, nonlocal discretization of the second
order elliptic equation.  The main observation is that, under suitable
assumptions, there is a spectral equivalence between a local operator arising
from an interior penalty method studied by Arnold and the above mentioned
nonlocal operator.  Hence, any reasonable preconditioner for the interior
penalty operator is also a suitable preconditioner for the corresponding mixed
system.  As an example of this approach we will use the interior penalty
method to build additive Schwarz and multigrid preconditioners for the mixed
system.

SINTEF, P. O. Box 124 Blindern, N-0314 Oslo, Norway.  E-mail address:
Torgeir.Rusten@si.sintef.no

Center of Informatics and Computer Technology, Bulgarian A cademy of Sciences,
"Acad. G. Bontchev" street, Block 25 A, 1113 Sofia, Bulgaria E-mail address:
panayot@bgearn.bitnet

Department of Informatics, University of Oslo, P. O. Box 1080 Blindern, N-0316
Oslo, Norway.  E-mail address:  Ragnar.Winther@ifi.uio.no

1991 Mathematics Subject Classification.  65F10, 65N20, 65N30.

Key words and phrases.  second order elliptic problems, mixed finite elements,
interior penalty method, domain decomposition.

The work of all authors was partially supported by The Research Council of
Norway (NFR), program no. 100998/420 and STP .29643.  The work of the second
author was also partially supported by Bulgarian Ministry for Science and
Education under grant MM-91#78.