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\(H\)-\(p\) clouds – an \(h\)-\(p\) meshless method. (English) Zbl 0869.65069
The so-called meshless methods (unlike the finite element or the finite difference methods) can dramatically simplify the large scale simulation of physical phenomena. Among the first methods of these types have been the methods due to T. Liszka [Int. J. Numer. Methods Eng. 20, 1599-1612 (1984; Zbl 0544.65006)] or to T. Belytschko, Y. Y. Lu, L. Gu [ibid. 37, No. 2, 229-256 (1994; Zbl 0796.73077)], based on ideas of P. Lancaster, K. Salkauskas [Math. Comp. 37, 141-158 (1981; Zbl 0469.41005)].
The article presents a new family of meshless methods for the solution of boundary value problems. The method uses radial basis functions of varying size of supports. Namely, the definition of a class of functions, to be used as trial and test functions for Galerkin approximations, is based on a partition of unity, connected with a scattered set of nodes. The idea behind that definition is the hierarchical addition of appropriate elements to the partition functions, that the resulting set represents (through linear combinations) polynomials of certain higher degree.
- Numerical experiments with the new technique in one or two dimensions show very promising results: Evidently the recent paper is a part of a running discussion. One of its essential aims is to bring more stringency (and this means: more analysis) into that domain of numerical mathematics.

MSC:
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J20 Variational methods for second-order elliptic equations
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