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On the choice of source points in the method of fundamental solutions. (English) Zbl 1244.65216
Summary: The method of fundamental solutions (MFS) may be seen as one of the simplest methods for solving boundary value problems for some linear partial differential equations (PDEs). It is a meshfree method that may present remarkable results with a small computational effort. The meshfree feature is particularly attractive when we need to change the shape of the domain, which occurs, for instance, in shape optimization and inverse problems. The MFS may be viewed as a Trefftz method, where the approximations have the advantage of verifying the linear PDE, and therefore we may bound the inner error from the boundary error, in well-posed problems. A main counterpart for these global numerical methods, that avoid meshes, are the associated linear systems with dense and ill conditioned matrices. In these methods a sort of uncertainty principle occurs-we cannot get both accurate results and good conditioning-one of the two is lost. A specific feature of the MFS is some freedom in choosing the source points. This might lead to excellent results, but it may also lead to poor results, or even to impossible approximations. In this work we will discuss the choice of source points and propose a choice along the discrete normal direction with a possible local criterion to define the distance to the boundary. We will also address some extensions that connect the asymptotic MFS to other methods by choosing the sources on a circle/sphere far from the boundary. We also present a direct connection between the approximation based on radial basis functions (RBF) and the MFS approximation in a higher dimension. This increase in dimension was somehow already present in a previous work [C. J. S. Alves and C. S. Chen, Adv. Comput. Math., No. 1–2, 125–142 (2005; Zbl 1070.65119)], where the frequency was used as the extra dimension. The free parameters in RBF inverse multiquadrics 2D approximation correspond in fact to the source point distance to the boundary plane in a Laplace 3D setting. Some numerical simulations are presented to illustrate theoretical issues.

MSC:
65N80 Fundamental solutions, Green’s function methods, etc. for boundary value problems involving PDEs
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