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Fundamental solutions method for elliptic boundary value problems. (English) Zbl 0579.65121
Elliptic boundary value problems can be solved using the method of fundamental solutions, first proposed by Kupradze. The method assumes as an approximation to the problem a finite sum of terms each of which is an unknown constant times a fundamental solution of the corresponding operator with singularity located outside the domain of interest. The constants are determined by a (possibly overdetermined) system of linear algebraic equations with matrix A. In case of a square matrix A it is known [the reviewer, Math. Meth. Appl. Sci. 3, 364-392 (1981; Zbl 0485.65089)] that the condition of A swiftly deteriorates as the distance between the singularities and the boundary increases. The paper investigates the method and finds that accurate solutions of the problems can still be obtained even though the conditioning is bad. This is illustrated by several examples which show that the best results may be obtained when the condition number is large - in some cases it was of the order \(10^{16}\). The results are presented as error curves. It could have been easier to understand the figures if they had been provided with captions, and the identification of the curves had been more elaborate. A conclusion of the paper would be of interest - the results are surprising.
Reviewer: S.Christiansen

MSC:
65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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