Balancing Domain Decomposition for Mixed Finite Elements

L. C. Cowsar
Department of Computational and Applied Mathematics
Rice University
Houston, TX 77251-1892

J. Mandel
Center for Computational Mathematics
University of Colorado at Denver
Denver, CO 80217-3364

M. F. Wheeler
Department of Computational and Applied Mathematics
Rice University
Houston, TX 77251-1892

Abstract

The rate of convergence of the Balancing Domain Decomposition method applied
to mixed finite element discretization of second order equations is analyzed.
The Balancing Domain Decomposition method, introduced by Mandel in [24], is a
substructuring method that involves at each iteration the solution of a local
problem with Dirichlet data, a local problem with Neumann data, and a "coarse
grid" problem to propogate globally and to insure the consistency of the
Neumann problems.  It is shown that the condition number grows at worst like
the logarithm sqquared of the subdomain size to the element size, in both two
and three dimensions and for elements of arbitrary order.  The bounds are
uniform with respect to coefficient jumps of arbitrary size between
subdomains.  The key component of our analysis is the demonstration of the
equivalence of the norm induced by the bilinear form on the interface and the
H<sup>1/2</sup>-norm of the interpolant of the boundary data.  Computational
results from a message passing parallel implementation on a INTEL-Delta
machine demonstrate the scalability properties of the method and show almost
optimal linear observed speed-up for up to 64 processors.