×

zbMATH — the first resource for mathematics

Element-free Galerkin method: Convergence of the continuous and discontinuous shape functions. (English) Zbl 0918.73125
Summary: We consider numerical solutions of second-order elliptic partial differential equations, such as Laplace’s equation, or linear elasticity, in two-dimensional, non-convex domains by the element-free Galerkin method (EFG). This is a meshless method in which the shape functions are constructed by using weight functions of compact support. For non-convex domains, two approaches to the determination of whether a node affects approximation at a particular point are used, a contained path criterion, and the visibility criterion. We show that for non-convex domains the visibility criterion leads to discontinuous weight functions and discontinuous shape functions. The resulting approximation is no longer conforming, and its convergence must be established by inspection of the so-called consistency term. We show that the variant of the element-free Galerkin method which uses the discontinuous shape functions, is convergent, and that, in the practically important case of linear shape functions, the convergence rate is not affected by the discontinuities. The convergence of the discontinuous approximation is first established by the classical and generalized patch test. As these tests do not provide an estimate of the convergence rate, the rate of convergence in the energy norm is examined, for both the continuous and discontinuous EFG shape functions and for smooth and non-smooth solutions by a direct inspection of the error terms.

MSC:
74S05 Finite element methods applied to problems in solid mechanics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Duarte, C.A., A review of some meshless methods to solve partial differential equations, ()
[2] Liszka, T.; Orkisz, J., The finite difference method at arbitrary irregular grids and its applications in applied mechanics, Comput. struct., 11, 83-95, (1980) · Zbl 0427.73077
[3] Monaghan, J.J., An introduction to SPH, Comput. phys. commun., 48, 89-96, (1982) · Zbl 0673.76089
[4] 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
[5] Kansa, E.J., Multiquadrics — a scattered data approximation scheme with applications to computational fluid dynamics: I. surface approximations and partial derivative estimates, Comput. math. applic., 19, 127-145, (1990) · Zbl 0692.76003
[6] Kansa, E.J., Multiquadrics — a scattered data approximation scheme with applications to computational fluid dynamics: II. solutions to parabolic, hyperbolic and elliptic partial differential equations, Comput. math. applic., 19, 147-161, (1990) · Zbl 0850.76048
[7] Belytschko, T.; Lu, Y.Y.; Gu, L., Element-free Galerkin methods, Int. J. numer. methods engrg., 37, 229-256, (1994) · Zbl 0796.73077
[8] Qian, S.; Weiss, J., Wavelet and the numerical solution of partial differential equations, J. comput. phys., 106, 155-175, (1993) · Zbl 0771.65072
[9] Duarte, C.A.; Oden, J.T., hp clouds — A meshless method to solve boundary-value problems, () · Zbl 0976.74071
[10] Liu, W.K.; Jun, Sukky; Li, S.; Adee, J.; Belytschko, T., Reproducing kernel particle methods for structural dynamics, Int. J. numer. methods engrg., 38, 1655-1679, (1995) · Zbl 0840.73078
[11] Babuška, I.; Melenk, J.M., The partition of unity finite element method, () · Zbl 0949.65117
[12] Babuška, I.; Melenk, J.M., The partition of unity finite element method, Int. J. numer. methods engrg., 40, 727-758, (1997) · Zbl 0949.65117
[13] Krysl, P.; Belytschko, T., Analysis of thin shells by the element-free Galerkin method, Int. J. solids struct., 33, 3057-3080, (1996) · Zbl 0929.74126
[14] Krysl, P.; Belytschko, T., Analysis of thin plates by the element-free Galerkin method, Comput. mech., 17, 26-35, (1996) · Zbl 0841.73064
[15] Cleveland, W.S., Visualizing data, (1993), AT & T bell Laboratories Murray Hill, NJ
[16] Lancaster, P.; Salkauskas, K., Curve and surface Fitting: an introduction, (1986), Academic Press London, Orlando · Zbl 0649.65012
[17] Belytschko, T.; Organ, D.; Krongauz, Y., A coupled finite element-element-free Galerkin method, Comput. mech., 17, 186-195, (1995) · Zbl 0840.73058
[18] Krongauz, Y.; Belytschko, T., Enforcement of essential boundary conditions in meshless approximations using finite elements, Comput, methods appl. mech. engrg., 131, 133-145, (1996) · Zbl 0881.65098
[19] Stummel, F., The generalized patch test, SIAM J. numer. anal., 3, 449-471, (1979) · Zbl 0418.65058
[20] Strang, W.G.; Fix, G.J., An analysis of the finite element method, (1973), Prentice-Hall Englewood Cliffs, NJ · Zbl 0278.65116
[21] Lancaster, P.; Salkauskas, K., Surfaces generated by moving least squares methods, Math. comput., 37, 141-158, (1981) · Zbl 0469.41005
[22] Organ, D.J.; Fleming, M.; Terry, T.; Belytschko, T., Continuous meshless approximations for nonconvex bodies by diffraction and transparency, Comput. mech., 18, 225-235, (1996) · Zbl 0864.73076
[23] Ciarlet, P.G., Basic error estimates for elliptic problem, (), Part 1 · Zbl 0875.65086
[24] Brenner, S.C.; Scott, L.R., The mathematical theory of finite element methods, (1994), Springer-Verlag New York · Zbl 0804.65101
[25] Liu, W.K.; Li, S.; Belytschko, T., Reproducing least square kernel Galerkin method. (i) methodology and convergence, Comput. methods appl. mech. engrg., 143, 113-154, (1997) · Zbl 0883.65088
[26] Belytschko, T.; Krongauz, Y.; Fleming, M.; Organ, D.; Liu, W.K., Smoothing and accelerated computations in the element-free Galerkin method, J. comput. appl. math., 74, 111-126, (1996) · Zbl 0862.73058
[27] Grisvard, P., Elliptic problems in nonsmooth domain, (1985), Pitman Advanced Pub. Program Boston · Zbl 0695.35060
[28] Babuška, I.; Rosenzweig, M.B., A finite element scheme for domains with corners, Numer. math., 20, 1-21, (1972) · Zbl 0252.65084
[29] Babǔska, I.; Kellog, R.B.; Pitkäranta, J., Direct and inverse error estimates for finite element with mesh refinements, Numer. math., 33, 447-471, (1979) · Zbl 0423.65057
[30] Babuška, I.; Dorr, M.R., Error estimates for the combined h and p versions of the finite element method, Numer. math., 37, 257-277, (1981) · Zbl 0487.65058
[31] Babuška, I.; Miller, K., The post-processing approach in the finite element method — part II. calculation of stress intensity factors, Int. J. numer. methods engrg., 20, 1111-1129, (1984) · Zbl 0535.73053
[32] Bourlard, M.; Dauge, M.; Lubuma, M.S.; Nicaise, S., Coefficients of the singularities for elliptic boundary value problems on domains with conical points. III: finite element methods on polygonal domains, SIAM J. numer. anal., 29, 136-155, (1992) · Zbl 0794.35015
[33] Wahlbin, L.B., Local behavior in finite element methods, (), 353-522, Part 1 · Zbl 0875.65089
[34] Dautray, R.; Lions, J-L., Mathematical analysis and numerical methods for science and technology, (1988), Springer-Verlag Berlin, New York, Part 2
[35] Rektorys, K., Variational methods in mathematics, science, and engineering, (1980), D. Reidel, Pub. Co Dordrecht, Holland, Boston, USA · Zbl 0481.49002
[36] Szabo, B.; Babuška, I., Finite element analysis, (1991), Wiley New York
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.