A Nonoverlapping Domain Decomposition Method for Symm's Equation for Conformal Mapping

Abstract

Symm's equation is a first-kind integral equation for computing conformal maps of simply connected regions. The package CONFPACK solves Symm's equation by an indirect boundary element method using an accurate corner representation. This solution technique is extended here to include nonoverlapping domain decomposition. Degrees of freedom are introduced on one or more interfaces and different unknowns are used, leading to a system of second-kind equations. The corresponding linear system can be expressed in Schur complement form. The accurate treatment of corners is preserved in the new formulation. The results of serial and parallel MATLAB implementations of the new algorithm show significant speedups as the number of unknowns grows.

A Study of Ray-Chaotic Cylindrical Scatterers

“…Note that, in view of the azimuthal periodicity, the power series in (37) contains only terms associated with integer multiples of . By substituting in (36) the inverse mapping , we obtain (38) By straightforward application of the approach in [24], which exploits the asymptotic (large-order) properties of Faber polynomials [36], it can be shown that the summations on the righthand side of (38) asymptotically approach a Fourier series on the unit circle in the complex -plane. Hence, truncating to the summations in (38), the point-matching problem is reduced to a truncated-Fourier-series approximation of a known function on the unit circle of an auxiliary complex plane, for which it is well known that uniform convergence is guaranteed provided that the sampling points are chosen as equispaced on the unit-circle [37].…”

“…The conformal mapping can be determined analytically only for a few boundary geometries [24]. For more general geometries, a standard numerical algorithm is based on momentmethod-type solution of the related Symm's integral equation [38]. In our implementation, we follow a different numerical approach, based on the approximation of via a truncated Laurent-series template (39) where and are unknown coefficients to be computed by enforcing that the transformation maps the unit-circle onto the contour in the -plane.…”

Finite Element Formulation in Flat Coordinate Spaces to Solve Elliptic Problems on General Closed Riemannian Manifolds

“…For example, the optimized Schwarz nonoverlapping method [5,24,27,9,10,16,17,27,28,32] can be easily implemented. The same idea appeared in [12], which computed a global conformal map by a nonoverlapping domain decomposition method. As nonoverlapping charts may not be easily constructed in general (cf.…”

Unified analysis on Petrov-Galerkin method into Symm's integral of the first kind

“…Symm's integral equation of the first kind plays an important role in constructing the conformal mappings (See [8,9,21]) and solving the Dirichlet and Neumann problem for Laplace equations. The general model is formulated as follows.…”