An alternative coarse space for irregular subdomains and an overlapping Schwarz algorithm for scalar elliptic problems in the plane?

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/Δ) was established assuming only that the subdomains are John domains; here H/Δ 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/Δ) 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.