×

zbMATH — the first resource for mathematics

Analysis of the complex moving least squares approximation and the associated element-free Galerkin method. (English) Zbl 1446.65176
Summary: The complex moving least squares approximation is an efficient method to construct approximation functions in meshless methods. This paper begins by analyzing properties, stability and error of the approximation. To overcome the inherent instability, a stabilized approximation is also developed and analyzed. The complex element-free Galerkin method is a meshless method combined with the use of the complex moving least squares approximation. Application of the complex element-free Galerkin method to linear and nonlinear time-dependent problems is then given. Error estimates of the complex element-free Galerkin method are derived theoretically. Numerical examples involving function fitting and solitons are finally provided to show the accuracy and efficiency of the proposed methods.

MSC:
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
74S05 Finite element methods applied to problems in solid mechanics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Lancaster, P.; Salkauskas, K., Surface generated by moving least squares methods, Math. Comput., 37, 141-158 (1981) · Zbl 0469.41005
[2] Belytschko, T.; Lu, Y. Y.; Gu, L., Element-free Galerkin methods, Int. J. Numer. Methods Eng., 37, 229-256 (1994) · Zbl 0796.73077
[3] Atluri, S. N., The Meshless Method (MLPG) for Domain & BIE Discretizations (2004), Tech. Science Press: Tech. Science Press California · Zbl 1105.65107
[4] Mukherjee, Y. X.; Mukherjee, S., The boundary node method for potential problems, Int. J. Numer. Methods Eng., 40, 797-815 (1997) · Zbl 0885.65124
[5] Li, X. L.; Zhu, J. L., A Galerkin boundary node method and its convergence analysis, J. Comput. Appl. Math., 230, 314-328 (2009) · Zbl 1189.65291
[6] Liew, K. M.; Feng, C.; Cheng, Y. M.; Kitipornchai, S., Complex variable moving least-squares method: a meshless approximation technique, Int. J. Numer. Methods Eng., 70, 46-70 (2007) · Zbl 1194.74554
[7] Shivanian, E., Meshless local Petrov-Galerkin (MLPG) method for three-dimensional nonlinear wave equations via moving least squares approximation, Eng. Anal. Bound. Elem., 50, 249-257 (2015) · Zbl 1403.65076
[8] Li, D. M.; Bai, F. N.; Cheng, Y. M.; Liew, K. M., A novel complex variable element-free Galerkin method for two-dimensional large deformation problems, Comput. Methods Appl. Mech. Eng., 233-236, 1-10 (2012) · Zbl 1253.74106
[9] Liew, K. M.; Cheng, Y. M., Complex variable boundary element-free method for two dimensional elastodynamic problems, Comput. Methods Appl. Mech. Eng., 198, 3925-3933 (2009) · Zbl 1231.74502
[10] Peng, M. J.; Li, D. M.; Cheng, Y. M., The complex variable element-free Galerkin (CVEFG) method for elasto-plasticity problems, Eng. Struct., 33, 127-135 (2011)
[11] Dai, B. D.; Wang, Q. F.; Zhang, W. W.; Wang, L. H., The complex variable meshless local Petrov-Galerkin method for elastodynamic problems, Appl. Math. Comput., 243, 311-321 (2014) · Zbl 1335.74054
[12] Weng, Y. J.; Zhang, Z.; Cheng, Y. M., The complex variable reproducing kernel particle method for two-dimensional inverse heat conduction problems, Eng. Anal. Bound. Elem., 44, 36-44 (2014) · Zbl 1297.65117
[13] Zuppa, C., Error estimates for moving least-square approximations, Bull. Braz. Math. Soc. New Ser., 34, 231-249 (2003) · Zbl 1056.41007
[14] Duarte, C. A.; Oden, J. T., H-p clouds—an h-p meshless method, Numer. Methods Part. Differ. Equ., 12, 673-705 (1996) · Zbl 0869.65069
[15] Wendland, H., Local polynomial reproduction and moving least squares approximation, IMA J. Numer. Anal., 21, 285-300 (2001) · Zbl 0976.65013
[16] Mirzaei, D., Analysis of moving least squares approximation revisited, J. Comput. Appl. Math., 282, 237-250 (2015) · Zbl 1309.65137
[17] Han, W. M.; Meng, X. P., Error analysis of the reproducing kernel particle method, Comput. Methods Appl. Mech. Eng., 190, 6157-6181 (2001) · Zbl 0992.65119
[18] Mirzaei, D.; Schaback, R.; Dehghan, M., On generalized moving least squares and diffuse derivatives, IMA. J. Numer. Anal., 32, 983-1000 (2012) · Zbl 1252.65037
[19] Cheng, R. J.; Cheng, Y. M., Error estimates for the finite point method, Appl. Numer. Math., 58, 884-898 (2008) · Zbl 1145.65086
[20] Li, X. L., Error estimates for the moving least-square approximation and the element-free Galerkin method in n-dimensional spaces, Appl. Numer. Math., 99, 77-97 (2016) · Zbl 1329.65274
[21] Li, X. L.; Zhang, S. G.; Wang, Y.; Chen, H., Analysis and application of the element-free Galerkin method for nonlinear Sine-Gordon and generalized Sinh-Gordon equations, Comput. Math. Appl., 71, 1655-1678 (2016)
[22] Burden, R. L.; Faires, J. D., Numerical Analysis (2011), Cengage Learning: Cengage Learning Boston
[23] Li, X. L., A meshless interpolating Galerkin boundary node method for stokes flows, Eng. Anal. Bound. Elem., 51, 112-122 (2015) · Zbl 1403.76090
[24] Davydov, O.; Zeilfelder, F., Scattered data fitting by direct extension of local polynomials to bivariate splines, Adv. Comput. Math., 21, 223-271 (2004) · Zbl 1065.41017
[25] Argyris, J.; Haase, M.; Heinrich, J. C., Finite element approximation to two- dimensional Sine-Gordon solitons, Comput. Methods Appl. Mech. Eng., 86, 1-26 (1991) · Zbl 0762.65073
[26] Dehghan, M.; Shokri, A., A numerical method for solution of the two-dimensional Sine-Gordon equation using the radial basis functions, Math. Comput. Simul., 79, 700-715 (2008) · Zbl 1155.65379
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.