zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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:
74S05Finite element methods in solid mechanics
65N30Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE)
WorldCat.org
Full Text: DOI
References:
[1] Duarte, C. A.: A review of some meshless methods to solve partial differential equations. Technical report 95-06 (1995)
[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. Technical report 95-05 (1995)
[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. Technical report BN-1185 (1995)
[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)
[16] Lancaster, P.; Salkauskas, K.: Curve and surface Fitting: an introduction. (1986) · 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) · Zbl 0356.65096
[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. Handbook of numerical analysis (1991) · Zbl 0875.65086
[24] Brenner, S. C.; Scott, L. R.: The mathematical theory of finite element methods. (1994) · 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) · 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] Babu\check{}ska, I.; Kellog, R. B.; Pitkäranta, J.: Direct and inverse error estimates for finite element with mesh refinements. Numer. math. 33, 447-471 (1979)
[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. Handbook of numerical analysis, 353-522 (1991) · Zbl 0875.65089
[34] Dautray, R.; Lions, J-L.: Mathematical analysis and numerical methods for science and technology. (1988) · Zbl 0664.47001
[35] Rektorys, K.: Variational methods in mathematics, science, and engineering. (1980) · Zbl 0481.49002
[36] Szabo, B.; Babuška, I.: Finite element analysis. (1991)