Abstract
A numerical procedure will be described and analyzed for the solution of various boundary value problems for Laplace’s equation
in three dimensional regions. This paper will discuss both the Dirichlet and Neumann problems, on regions both interior and exterior to a smooth simple closed surface S. Each such problem is reduced to an equivalent integral equation over S by representing the solution u as a single or double layer potential; and the resulting integral equation is then reformulated as an integral equation on the unit sphere U in ℝ3, using a simple change of variables. This final equation over U is solved numerically using Galerkin’s method, with spherical harmonics as the basis functions. The resulting numerical method converges rapidly, although great care must be taken to evaluate the Galerkin coefficients as efficiently as possible.
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Atkinson, K.E. (1980). The Numerical Solution of Laplace’s Equation in Three Dimensions—II. In: Albrecht, J., Collatz, L. (eds) Numerical Treatment of Integral Equations / Numerische Behandlung von Integralgleichungen. International Series of Numerical Mathematics / International Schriftenreihe zur Numerischen Mathematik / Série internationale d’Analyse numérique, vol 53. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-6314-8_1
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DOI: https://doi.org/10.1007/978-3-0348-6314-8_1
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