in Domain Decomposition Preconditioners

Johannes Kepler University

Institut für Mathematik

A-4040 Linz, Altenbergerstrasse 69, Austria

The paper presents a cheap technique for the approximation of the harmonic
extension from the boundary into the interior of a domain with respect to a
given differential operator. The new extension operator is based on the
hierarchical splitting of the given f.e. space together with smoothing sweeps
and an exact discrete harmonic extension on the lowest level and will be used
as a component in a domain decomposition (DD) preconditioner. In combination
with an additional algorithmical improvement of this DD-preconditioner
solution times faster then the previously studied were achieved for the
preconditioned parallelized cg-method. The analysis of the new extension
operator gives the result that in the 2D-case O(ln(ln(h^{-1})))
smoothing sweeps per level are sufficient to achieve an h-independent behavior
of the preconditioned system provided that there exists a spectrally
equivalent preconditioner for the modified Schur complement with spectral
equivalence constants independent of h.

Keywords: Boundary value problems, Finite element method, Domain decomposition, Preconditioning, Parallel iterative solvers.

Contributed November 27, 1997.