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Applied Math Seminar
Locally-corrected spectral boundary integral methods
for elliptic systems
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Classical potential theory converts linear constant-coefficient elliptic problems in complex domains into integral equations on interfaces, and generates robust, efficient numerical methods. The conversion is usually carried out for a particular situation such as the Poisson equation in dimension 2, and the efficiency of the resulting methods then depends on detailed analysis of the appropriate special functions. We present a general conversion scheme which leads naturally to a fast general algorithm: arbitrary elliptic problems in arbitrary dimension are converted to first-order systems, a periodic fundamental solution is mollified for convergence, and the mollification is locally corrected via Ewald summation. Local linear algebra and the elementary theory of distributions yield a simple boundary integral equation. With the aid of a new nonequidistant fast Fourier transform for piecewise polynomial functions, the resulting numerical methods provide highly accurate solutions to general elliptic systems in complex domains. |