# Small Coarse Spaces for Overlapping Schwarz Algorithms with Irregular Subdomains

## Abstract

Methods are developed for automatically constructing small coarse spaces of low dimension for domain decomposition algorithms for problems in three dimensions. These constructions use equivalence classes of nodes on the interface between the subdomains into which the domain of a given elliptic problem has been subdivided, e.g., by a mesh partitioner; these equivalence classes already play a central role in the design, analysis, and programming of many domain decomposition algorithms. The coarse space elements are well defined even for irregular subdomains, are continuous, and well suited for use in two-level or multi-level preconditioners such as overlapping Schwarz algorithms. Significant reductions in the coarse space dimension can be achieved while not sacrificing the favorable condition number estimates for larger coarse spaces previously developed. The condition number estimates depend primarily on the Lipschitz parameters of the subdomains.

## Notes

### Acknowledgements

The work of the first author was supported in part by the National Science Foundation Grant DMS-1522736.

Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia LLC, a wholly owned subsidiary of Honeywell International Inc. for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525.

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