On the use of rational-function fitting methods for the solution of 2D Laplace boundary-value problems

• Main Authors: Hochman, Amit (Contributor), Leviatan, Yehuda (Author), White, Jacob K. (Contributor)
• Other Authors: Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science (Contributor), Massachusetts Institute of Technology. Research Laboratory of Electronics (Contributor)
• Format: Article
• Language: English
• Published: Elsevier B.V., 2014-10-17T18:18:17Z.
• Subjects: Article
• Online Access: Get fulltext

A computational scheme for solving 2D Laplace boundary-value problems using rational functions as the basis functions is described. The scheme belongs to the class of desingularized methods, for which the location of singularities and testing points is a major issue that is addressed by the proposed scheme, in the context he 2D Laplace equation. Well-established rational-function fitting techniques are used to set the poles, while residues are determined by enforcing the boundary conditions in the least-squares sense at the nodes of rational Gauss-Chebyshev quadrature rules. Numerical results show that errors approaching the machine epsilon can be obtained for sharp and almost sharp corners, nearly-touching boundaries, and almost-singular boundary data. We show various examples of these cases in which the method yields compact solutions, requiring fewer basis functions than the Nyström method, for the same accuracy. A scheme for solving fairly large-scale problems is also presented.
Technion, Israel Institute of Technology. Advanced Circuit Research Center
Singapore-MIT Alliance Computational Engineering Programme
USC Viterbi School of Engineering (Postdoctoral Fellowship)