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A spline interpolation method for solving boundary value problems of potential theory from discretely given data. (English) Zbl 0659.65111

The boundary value problem of the title is the exterior Robin problem for Laplace’s equation in three dimensions. It is shown that in an appropriate Hilbert space a reproducing kernel exists. This is constructed from spherical harmonics; a section is devoted to a summary of the required properties of these functions.
The representers of linear (evaluation) functionals associated with the boundary value problem can be written in terms of this kernel function. The author calls linear combinations of the representers ‘harmonic splines’. These are used to provide approximations to the solution of the problem. These approximations, which are smooth in a well defined sense, are found by interpolation. The interpolation problem is shown to be well posed, and that the approximations converge to the solution.
There is a discussion of theoretical and computaional aspects of the method. A list of reproducing kernels which can be written in closed form is given. Some numerical examples are presented when the bounding surface are either ellipsoids or derived from an oval of Cassini.
Reviewer: D.Kershaw

MSC:

65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
31B20 Boundary value and inverse problems for harmonic functions in higher dimensions

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References:

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