Domain Decomposition Methods for Conformal Mapping and Eigenvalue Problems
Driscoll, Tobin Allen
Domain decomposition is widely used in the numerical solution of elliptic boundary value problems. It is appealing in part because of improved efficiency and straightforward parallelization. Conceptually, domain decomposition often exploits a natural feature of elliptic problems: data at one point may have an exceedingly weak influence on the solution at a far-removed point. The application of domain decomposition to other elliptic problems is less fully developed. One such area is numerical conformal mapping. We introduce the SC Toolbox for Matlab, an interactive graphical software package for the Schwarz-Christoffel mapping of polygons. The Toolbox can be used for interior and exterior mapping from several fundamental domains. Elongations in the polygon lead to crowding, in which the preimages of affected vertices are exponentially close together. Such regions are candidates for decomposition. We describe CRDT, an overlapping subdomain method developed with Vavasis for numerical Schwarz-Christoffel mapping. The method uses Delaunay triangulation to decompose the polygon into overlapping quadrilaterals, which in turn define cross-ratios that form the basis of a nonlinear system. Each quadrilateral induces an embedding of the prevertices so that locally, the map can be computed accurately. Apparently CRDT can deal with any degree of crowding, as is demonstrated by examples. Another application in conformal mapping is in Symm's integral equation. An important feature of existing software for Symm's equation is the efficient treatment of corner singularities. Careful generalization to multiple domains allows this treatment to be preserved and extended. An onoverlapping formulation leads to a linear system that is ideal for Schur complementation. The resulting method asymptotically requires a fraction of the single-domain work and is easily parallelized. We also consider a domain decomposition algorithm for the Laplace eigenvalue problem on polygons. This method, an improvement on one described by Descloux and Tolley, searches for the matching of Fourier-Bessel expansions at each corner to locate eigenvalues. We apply the algorithm to the "isospectral drums" discovered by Gordon, Webb, and Wolpert to find 25 eigenvalues to 12 digits. The method is far more accurate and efficient than standard methods for this problem.
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