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Generalizing the finite element method: Diffuse approximation and diffuse elements. (English) Zbl 0764.65068
A new approximation method, called diffuse approximation, is presented. It is based on a local weighted least squares polynomial fitting. It provides local but continuous approximations of functions and of their successive derivatives. This new method provides better gradients of the unknown functions than the finite element method. It requires sets of discretization points (or nodes) and no explicit elements. Some numerical tests are included.
Reviewer: K.Najzar (Praha)

MSC:
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
35J25 Boundary value problems for second-order elliptic equations
34B05 Linear boundary value problems for ordinary differential equations
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[1] Nayroles, B.; Touzot, G.; Villon, P. (1991a): La méthode des éléments diffus. C.R. Acad. Sci. Paris, t. 313, Série II, pp 293-296 · Zbl 0725.73085
[2] Nayroles, B.; Touzot, G.; Villon, P. (1991b): L’approximation diffuse. C.R. Acad. Sci. Paris, t. 313, Série II, pp 133-138 · Zbl 0725.73085
[3] Nayroles, B.; Touzot, G.; Villon, P. (1991c): Nuages de Points et Approximation diffuse. Séminaire d’analyse convexe, Exposé no 16, Université de Montpellier II · Zbl 0829.65123
[4] Nayroles, B.; Touzot, G.: Using diffuse approximation for optimizing antisound sources location. (submitted to J.S.V.) · Zbl 1064.76608
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