Authors: Clark R. Dohrmann and Olof B. Widlund

Title: An Alternative Coarse Space for Irregular Subdomains and an Overlapping Schwarz Algorithm

Abstract

In earlier work on domain decomposition methods for elliptic problems in the plane,
an assumption that each subdomain is triangular, or a union of
a few coarse triangles, has often been made. This is
similar to what is required in geometric multigrid theory
and is unrealistic if the subdomains are produced by a mesh partitioner.
In an earlier paper, coauthored with Axel Klawonn, the authors introduced
a coarse subspace for an overlapping Schwarz method
with one degree of freedom for each subdomain
vertex and one for each subdomain edge. A condition number
bound proportional to $(1+\log(H/h))^2(1+H/\delta)$
was established assuming only that the subdomains are
John domains; here $H/\delta$ measures the relative overlap between
neighboring subdomains and $H/h$ the maximum
number of elements across individual subdomains. We were
also able to relate the rate of convergence to a parameter
in an isoperimetric inequality for the subdomains into which
the domain of the problem has been partitioned.

In this paper, the dimension of the coarse subspace is decreased
by using only one degree of freedom for each subdomain
vertex; if all subdomains have three edges, this leads to a reduction
of the dimension of the coarse subspace by approximately a factor four.
In addition, the condition number bound is shown to be proportional to
$(1+\log(H/h))(1+H/\delta)$ under a quite mild assumption on the
relative length of adjacent subdomain edges.

In this study, the subdomains are assumed to be uniform in
the sense of Peter Jones. As in our earlier work, the results are insensitive
to arbitrary large jumps in the coefficients of the elliptic problem across
the interface between the subdomains.

Numerical results are presented which confirm the
theory and demonstrate the usefulness of the algorithm for a variety
of mesh decompositions and distributions of material properties. It is
also shown that the new algorithm often converges faster than the older
one in spite of the fact that the dimension of the coarse space has been
decreased considerably.