zbMATH — the first resource for mathematics

Improvements of generalized finite difference method and comparison with other meshless method. (English) Zbl 1046.65085
Summary: One of the most universal and effective methods, in wide use today, for approximately solving equations of mathematical physics is the finite difference (FD) method. An evolution of the FD method has been the development of the generalized finite difference (GFD) method, which can be applied over general or irregular clouds of points. The main drawback of the GFD method is the possibility of obtaining ill-conditioned stars of nodes.
In this paper a procedure is given that can easily assure the quality of numerical results by obtaining the residual at each point. The possibility of employing the GFD method over adaptive clouds of points increasing progressively the number of nodes is explored, giving in this paper a condition to be accomplished to employ the GFD method with more efficiency.
Also, in this paper, the GFD method is compared with another meshless method the, so-called, element free Galerkin method (EFG). The EFG method with linear approximation and penalty functions to treat the essential boundary condition is used in this paper. Both methods are compared for solving Laplace equation.

MSC:
 65N06 Finite difference methods for boundary value problems involving PDEs 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
Full Text:
References:
 [1] Lucy, L.B., A numerical approach to the testing of the fission hypothesis, Astron. J., 82, 12, 1013-1024, (1977) [2] Perrone, N.; Kao, R., A general finite difference method for arbitrary meshes, Comput. struct., 5, 45-58, (1975) [3] Jensen, P.S., Finite difference techniques for variable grids, Comput. struct., 2, 17-29, (1972) [4] Nay, R.A.; Utku, S., An alternative for the finite element method, Variational meth. eng., 3, 62-74, (1973) · Zbl 0312.73087 [5] Liszka, T.; Orkisz, J., The finite difference method at arbitrary irregular grids and its application in applied mechanics, Comput. struct., 11, 83-95, (1980) · Zbl 0427.73077 [6] Liszka, T., An interpolation method for an irregular net of nodes, Int. J. numer. meth. eng., 20, 1599-1612, (1984) · Zbl 0544.65006 [7] Lancaster, P.; Salkauskas, K., Surfaces generated by moving least squares methods, Math. comput., 37, 141-158, (1981) · Zbl 0469.41005 [8] Orkisz, J., Mesless finite difference method. I. basic approach, in computational mechanics. new trends and applications, () [9] Benito, J.J.; Ureña, F.; Gavete, L., Influence of several factors in the generalized finite difference method, Appl. math. modell., 25, 12, 1039-1053, (2001) · Zbl 0994.65111 [10] Nayroles, B.; Touzot, G.; Villon, P., Generalizing the finite element method:diffuse approximation and diffuse elements, Computat. mech., 10, 307-318, (1992) · Zbl 0764.65068 [11] Belytschko, T.; Lu, Y.Y.; Gu, L., Element-free Galerkin methods, Int. J. numer. meth. eng., 37, 229-256, (1994) · Zbl 0796.73077 [12] Gavete, L.; Benito, J.J.; Falcon, S.; Ruiz, A., Implementation of essential boundary conditions in a meshless method, Commun. numer. meth. eng., 16, 409-421, (2000) · Zbl 0956.65104 [13] Gavete, L.; Benito, J.J.; Falcon, S.; Ruiz, A., Penalty functions in constrained variational principles for element free Galerkin method, Eur. J. mech. (a) solids, 19, 699-720, (2000) · Zbl 0992.74078 [14] Liu, W.K.; Jun, S.; Li, S.; Adee, J.; Belytschko, T., Reproducing kernel particle methods for structural dynamics, Int. J. numer. meth. eng., 38, 1655-1679, (1995) · Zbl 0840.73078 [15] Oñate, E.; Idelsohn, S.; Zienkiewicz, O.C.; Taylor, R.L., A finite point method in computational mechanics. aplications to convective transport and fluid flow, Int. J. numer. meth. eng., 39, 3839-3866, (1996) · Zbl 0884.76068 [16] Duarte, A.; Oden, J.T., H-P cloud–an h-p meshless method, Numer. meth. partial differen. equat., 12, 673-705, (1996) · Zbl 0869.65069 [17] Babuska, I.; Melenk, J.M., The partition of unity method, Int. J. numer. meth. eng., 40, 727-758, (1997) · Zbl 0949.65117 [18] Syczewski, M.; Tribillo, R., Singularities of sets used in the mesh method, Comput. struct., 14, 5-6, 509-511, (1981)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.