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An \(h\)-adaptive method in the generalized finite differences. (English) Zbl 1024.65099
Summary: This paper describes an \(h\)-adaptive method in generalized finite difference (GFD) to solve second-order partial differential equations. These equations representing the behaviour of many physical processes. The explicit difference formulae obtained make it possible to propose an a posteriori error indicator which serves as starting point for an \(h\)-adaptive method to improve the solution by selectively adding nodes to the domain.
This paper also analyses the influence of key parameters, as the number of nodes to add in each step or the minimum distance between nodes, through the analysis of the obtained solutions for different types of differential equations.

65N06 Finite difference methods for boundary value problems involving PDEs
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74S20 Finite difference methods applied to problems in solid mechanics
35G15 Boundary value problems for linear higher-order PDEs
65N15 Error bounds for boundary value problems involving PDEs
Full Text: DOI
[1] Babuska, I.; Melenk, J.M., The partion of unity method, Internat. J. numer. methods engrg., 40, 727-758, (1997) · Zbl 0949.65117
[2] Belytschko, T.; Lu, Y.Y.; Gu, L., Element free-Galerkin method, Internat. J. numer. methods engrg., 37, 229-256, (1994) · Zbl 0796.73077
[3] J.J. Benito, Métodos sin malla, trabajo de investigación presentado para el concurso de una plaza de Catedrático de Universidad, Madrid, Spain, 1998 (in Spanish)
[4] Benito, J.J.; Ureña, F.; Gavete, L., Influence of several factors in the generalized finite difference method, Appl. math. modelling, 25, 1039-1053, (2001) · Zbl 0994.65111
[5] Cendrowicz, J.; Tribillo, R., Variational approach to the static analysis of plates of an arbitrary shape, Arch. inz. ladowej, 3, 24, (1978), (in Polish)
[6] Duarte, A.; Oden, J.T., H-P cloud–an h-p meshless method, Numer. methods partial differential equations, 12, 673-705, (1996) · Zbl 0869.65069
[7] Forsythe, G.E.; Wasow, W.R., Finite-difference methods for partial differential equations, (1960), Wiley New York · Zbl 0099.11103
[8] Frey, W.H., Flexible finite-difference stencils from isoparametric finite elements, Internat. J. numer. methods engrg., 12, 229-235, (1978)
[9] Gavete, L.; Benito, J.J.; Falcón, S.; Ruiz, A., Implementation of essential boundary conditions in a meshless method, Commun. numer. methods engrg., 16, 409-421, (2000) · Zbl 0956.65104
[10] Jensen, P.S., Finite difference technique for variable grids, Comput. struct., 2, 17-29, (1972)
[11] Kaczkowski, Z.; Tribillo, R., A generalization of the finite difference method, Arch. inz. ladowej, 2, 21, 287-293, (1975), (in Polish)
[12] Lancaster, J.; Salkauskas, K., Surfaces generated by moving squares methods, Math. comput., 155, 37, 141-158, (1981) · Zbl 0469.41005
[13] Liszka, T., An interpolation method for an irregular net of nodes, Internat. J. numer. methods engrg., 20, 1599-1612, (1984) · Zbl 0544.65006
[14] Liszka, T.; Orkisz, J., The finite difference method at arbitrary irregular grids and its application in applied mechanics, (), 83-95 · Zbl 0427.73077
[15] Liu, W.K.; Jun, S.; Li, S.; Adee, J.; Belytschko, T., Reproducing kernel particle methods for structural dynamics, Internat. J. numer. methods engrg., 38, 1655-1679, (1995) · Zbl 0840.73078
[16] MacNeal, R.H., An asymmetrical finite difference network, Appl. math., 11, 295-310, (1953) · Zbl 0053.26304
[17] Nayroles, B.; Touzot, G.; Villon, P., Generalizing the finite element method: diffuse approximation and diffuse elements, Comput. mech., 10, 307-318, (1992) · Zbl 0764.65068
[18] Oñate, E.; Idelshon, S.; Zienkiewicz, D.C.; Taylor, R.L., A finite point method in computational mechanics. applications to conductive transport and fluid flow, Internat. J. numer. methods engrg., 39, 3839-3866, (1996) · Zbl 0884.76068
[19] J. Orkisz, Meshless finite difference method II. Adaptive approach, in: Idelshon, Oñate, Duorkin (Eds.), Computational Mechanics, IACM, CINME, 1998
[20] J. Orkisz, Meshless finite difference method I. Basic approach, in: Idelshon, Oñate, Duorkin (Eds.), Computational Mechanics, IACM, CINME, 1998
[21] Orkisz, J., Finite difference method (part III), (), 336-432
[22] Perrone, N.; Kao, R., A general finite difference method for arbitrary meshes, Computers struct., 5, 45-58, (1975)
[23] J. Szmelter, Z. Kurowski, A complete program for solving systems of linear partial differential equations in plain domains, (in Polish), in: Proceedings of Third Conference On Computational Methods In Structural Mechanics, Opole, Poland, 1977, pp. 237-247
[24] R. Tribillo, Application of algebraic structures to a generalized finite difference method, (in Polish) in: Politechnika Bialestocka, Zeszyt Naukowy nr9, Bialystok, 1976
[25] Wyatt, M.J.; Davies, G.; Snell, C., A new difference based finite element method, Instn. engrs., 59, 2, 395-409, (1975)
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