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Methods of fundamental solutions for harmonic and biharmonic boundary value problems. (English) Zbl 0913.65104
The idea of the method of fundamental solutions (a variant of the boundary element method) is to approximate the solution by fundamental solutions of the governing equation, with singularities located outside of the domain of solution. The second author of the present paper – jointly with co-authors – examined the application of the method of fundamental solutions to problems in mechanics, mostly governed by the potential and the bipotential equation, see [J. Comput. Phys. 69, 434-459 (1987; Zbl 0618.65108), ibid. 98, No. 1, 119-128 (1992; Zbl 0745.65075), IMA J. Numer. Anal. 9, No. 2, 231-242 (1989; Zbl 0676.65110), Numer. Meth. Partial Diff. Equations 8, No. 1, 1-9 (1992; Zbl 0760.65103)].
An obstacle in the use of that method is the nonlinear least squares minimization to find the superposition of fundamental solution satisfying the boundary conditions. The authors discuss some routines to overcome the computational difficulties, among them the MINPACK routines due to B. S. Garbow, K. E. Hillstrom and J. J. Moré [ACM Trans. Math. Software 7, 17-41 (1981; Zbl 0454.65049)]. There is a promising progress.

MSC:
65N38 Boundary element methods for boundary value problems involving PDEs
35E05 Fundamental solutions to PDEs and systems of PDEs with constant coefficients
31A10 Integral representations, integral operators, integral equations methods in two dimensions
31A30 Biharmonic, polyharmonic functions and equations, Poisson’s equation in two dimensions
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35J40 Boundary value problems for higher-order elliptic equations
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