On the use of rational-function fitting methods for the solution of 2D Laplace boundary-value problems
Author(s)Hochman, Amit; Leviatan, Yehuda; White, Jacob K.
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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.
DepartmentMassachusetts Institute of Technology. Department of Electrical Engineering and Computer Science; Massachusetts Institute of Technology. Research Laboratory of Electronics
Journal of Computational Physics
Hochman, Amit, Yehuda Leviatan, and Jacob K. White. “On the Use of Rational-Function Fitting Methods for the Solution of 2D Laplace Boundary-Value Problems.” Journal of Computational Physics 238 (April 2013): 337–358.