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The Delta-Trigonometric Method using the Single-Layer Potential Representation

Summary: The Delta-Trigonometric Method using
the Single-Layer Potential Representation
Raymond Sheng-Chieh Cheng Douglas N. Arnold*
Global Associates, Ltd. Department of Mathematics
1423 Powhatan Street University of Maryland
Alexandria, VA 22314 College Park, MD 20742
The Dirichlet problem for Laplace's equation is often solved by means of the single layer
potential representation, leading to a Fredholm integral equation of the first kind with loga-
rithmic kernel. We propose to solve this integral equation using a Petrov-Galerkin method
with trigonometric polynomials as test functions and, as trial functions, a span of delta
distributions centered at boundary points. The approximate solution to the boundary value
problem thus computed converges exponentially away from the boundary and algebraically
up to the boundary. We show that these convergence results hold even when the discretiza-
tion matrices are computed via numerical quadratures. Finally, we discuss our imple-
mentation of this method using the fast Fourier transform to compute the discretization
matrices, and present numerical experiments in order to confirm our theory and to ex-
amine the behavior of the method in cases where the theory doesn't apply due to lack of
1980 Mathematics Subject Classification: Primary: 65R20; Secondary: 65N30, 65E05, 45L10


Source: Arnold, Douglas N. - School of Mathematics, University of Minnesota


Collections: Mathematics