Error analysis in Sobolev spaces for the improved moving least-square approximation and the improved element-free Galerkin method. (English) Zbl 1410.65456
Summary: The improved moving least-square (IMLS) approximation is a method to form shape functions in meshless methods. For the application of IMLS-based meshless methods to the numerical solution of boundary value problems, it is fundamental to analyze error of the IMLS approximation in Sobolev spaces. This paper begins by discussing properties of the IMLS shape function. Under appropriate assumption on weight functions, error estimates for the IMLS approximation are then established in Sobolev spaces in multiple dimensions. The improved element-free Galerkin (IEFG) method is a typical meshless Galerkin method based on coupling the IMLS approximation and Galerkin weak form. Error analysis of the IEFG method is also provided. Numerical examples are finally presented to prove the theoretical error results.
MSC:
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
Keywords:
error analysis; Sobolev spaces; improved moving least-square approximation; improved element-free Galerkin method; meshless methodReferences:
| [1] | Lancaster, P.; Salkauskas, K., Surface generated by moving least squares methods, Math. Comput., 37, 141-158, (1981) · Zbl 0469.41005 |
| [2] | 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 |
| [3] | Belytschko, T.; Lu, Y. Y.; Gu, L., Element-free Galerkin methods, Int. J. Numer. Methods Eng., 37, 229-256, (1994) · Zbl 0796.73077 |
| [4] | Hajiazizi, M.; Bastan, P., The elastoplastic analysis of a tunnel using the EFG method: A comparison of the EFGM with FEM and FDM, Appl. Math. Comput., 234, 82-113, (2014) · Zbl 1302.74162 |
| [5] | Duarte, C. A.; Oden, J. T., H-p clouds—an h-p meshless method, Numer. Methods Partial Diff. Eq., 12, 673-705, (1996) · Zbl 0869.65069 |
| [6] | Atluri, S. N., The Meshless Method (MLPG) for Domain & BIE Discretizations, (2004), Tech. Science Press California · Zbl 1105.65107 |
| [7] | Mazzia, A.; Pini, G.; Sartoretto, F., Meshless techniques for anisotropic diffusion, Appl. Math. Comput., 236, 54-66, (2014) · Zbl 1337.65155 |
| [8] | 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 |
| [9] | nate, E. O.; Idelsohn, S.; Zienkiewicz, O. C.; Taylor, R. L., A finite point method in computational mechanics. applications to convective transport and fluid flow, Int. J. Numer. Meth ods Eng., 39, 3839-3866, (1996) · Zbl 0884.76068 |
| [10] | Cheng, R. J.; Zhang, L. W.; Liew, K. M., Modeling of biological population problems using the element-free KP-Ritz method, Appl. Math. Comput., 227, 274-290, (2014) · Zbl 1364.65193 |
| [11] | Mukherjee, Y. X.; Mukherjee, S., The boundary node method for potential problems, Int. J. Numer. Methods Eng., 40, 797-815, (1997) · Zbl 0885.65124 |
| [12] | 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 |
| [13] | Li, X. L., Meshless Galerkin algorithms for boundary integral equations with moving least square approximations, Appl. Numer. Math., 61, 1237-1256, (2011) · Zbl 1232.65160 |
| [14] | Li, X. L., Implementation of boundary conditions in BIEs-based meshless methods: A dual boundary node method, Eng. Anal. Bound. Elem., 41, 139-151, (2014) · Zbl 1297.65180 |
| [15] | Lu, Y. Y.; Belytschko, T.; Gu, L., A new implementation of the element free Galerkin method, Comput. Methods Appl. Mech. Eng., 113, 397-414, (1994) · Zbl 0847.73064 |
| [16] | Liew, K. M.; Cheng, Y.; Kitipornchai, S., Boundary element-free method (BEFM) and its application to two-dimensional elasticity problems, Int. J. Numer. Methods Eng., 65, 1310-1332, (2006) · Zbl 1147.74047 |
| [17] | Zhang, Z.; Liew, K. M.; Cheng, Y. M.; Li, Y. Y., Analyzing 2D fracture problems with the improved element-free Galerkin method, Eng. Anal. Bound. Elem., 32, 241-250, (2008) · Zbl 1244.74240 |
| [18] | Zhang, Z.; Liew, K. M.; Cheng, Y. M., Coupling of improved element-free Galerkin and boundary element methods for the 2D elasticity problems, Eng. Anal. Bound. Elem., 32, 100-107, (2008) · Zbl 1244.74204 |
| [19] | Zhang, Z.; Zhao, P.; Liew, K. M., Analyzing three-dimensional potential problems with the improved element-free Galerkin method., Comput. Mech., 44, 273-284, (2009) · Zbl 1171.65083 |
| [20] | Zhang, L. W.; Deng, Y. J.; Liew, K. M., An improved element-free Galerkin method for numerical modeling of the biological population problems, Eng. Anal. Bound. Elem., 40, 181-188, (2014) · Zbl 1297.65123 |
| [21] | Peng, M. J.; Li, R. X.; Cheng, Y. M., Analyzing three-dimensional viscoelasticity problems via the improved element-free Galerkin (IEFG) method, Eng. Anal. Bound. Elem., 40, 104-113, (2014) · Zbl 1297.74153 |
| [22] | Levin, D., The approximation power of moving least-squares, Math. Comp., 67, 1335-1754, (1998) |
| [23] | Armentano, G.; Durán, R., Error estimates for moving least square approximations, Appl. Numer. Math., 37, 397-416, (2001) · Zbl 0984.65096 |
| [24] | Armentano, G., Error estimates in Sobolev spaces for moving least square approximations, SIAM J. Numer. Anal., 39, 38-51, (2002) · Zbl 1001.65014 |
| [25] | Zuppa, C., Error estimates for moving least-square approximations, Bull. Braz. Math. Soc. (N.S.), 34, 231-249, (2003) · Zbl 1056.41007 |
| [26] | Cheng, R. J.; Cheng, Y. M., Error estimates for the finite point method, Appl. Numer. Math., 58, 884-898, (2008) · Zbl 1145.65086 |
| [27] | Cheng, R. J.; Cheng, Y. M., Error estimates of element-free Galerkin method for potential problems, Acta Phys. Sinica, 10, 6037-6046, (2008) · Zbl 1199.78009 |
| [28] | Zuppa, C., Good quality point sets and error estimates for moving least square approximations, Appl. Numer. Math., 47, 575-585, (2003) · Zbl 1040.65034 |
| [29] | Li, X. L., The meshless Galerkin boundary node method for Stokes problems in three dimensions, Int. J. Numer. Methods Eng., 88, 442-472, (2011) · Zbl 1242.76244 |
| [30] | Li, X. L., Adaptive meshless Galerkin boundary node methods for hypersingular integral equations, Appl. Math. Model., 36, 4952-4970, (2012) · Zbl 1252.65197 |
| [31] | Li, X. L.; Li, S., A meshless Galerkin method with moving least square approximations for infinite elastic solids, Chin. Phys. B, 22, 080204, (2013) |
| [32] | Li, X. L., Symmetric coupling of the meshless Galerkin boundary node and finite element methods for elasticity, Comput. Model. Eng. Sci., 97, 483-507, (2014) · Zbl 1356.74020 |
| [33] | Ren, H. P.; Pei, K. Y.; Wang, L. P., Error analysis for moving least squares approximation in 2D space, Appl. Math. Comput., 238, 527-546, (2014) · Zbl 1337.65157 |
| [34] | Ren, H. P.; Cheng, Y. M.; Zhang, W., An interpolating boundary element-free method (IBEFM) for elasticity problems, Sci. China Ser. G Phys. Mech. Astron., 53, 758-766, (2010) |
| [35] | Wang, J. F.; Sun, F. X.; Cheng, Y. M., An improved interpolating element-free Galerkin method with nonsingular weight function for two-dimensional potential problems, Chin. Phys. B, 21, 090204, (2012) |
| [36] | Li, X. L., An interpolating boundary element-free method for three-dimensional potential problems, Appl. Math. Model., (2015), http://dx.doi.org/10.1016/j.apm.2014.10.071 · Zbl 1443.65407 |
| [37] | 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 |
| [38] | Ren, H. P.; Cheng, J.; Huang, A. X., The complex variable interpolating moving least-squares method, Appl. Math. Comput., 219, 1724-1736, (2012) · Zbl 1291.65138 |
| [39] | Liu, W. K.; Li, S. F.; Belytschko, T., Moving least-square reproducing kernel methods (I) methodology and convergence, Comput. Methods Appl. Mech. Eng., 143, 113-154, (1997) · Zbl 0883.65088 |
| [40] | Wang, J. F.; Sun, F. X.; Cheng, Y. M.; Huang, A. X., Error estimates for the interpolating moving least-squares method, Appl. Math. Comput., 245, 321-342, (2014) · Zbl 1335.65018 |
| [41] | Salehi, R.; Dehghan, M., A moving least-square reproducing polynomial meshless method, Appl. Numer. Math., 69, 34-58, (2013) · Zbl 1284.65137 |
| [42] | Assari, P.; Adibi, H.; Dehghan, M., A meshless discrete Galerkin (MDG) method for the numerical solution of integral equations with logarithmic kernels, J. Comput. Appl. Math., 267, 160-181, (2014) · Zbl 1293.65166 |
| [43] | Zuppa, C., Jachson-type inequalities for h-p clouds and error estimates, Comput. Methods Appl. Mech. Eng., 194, 1875-1887, (2005) · Zbl 1097.65110 |
| [44] | Zhu, J. L.; Yuan, Z. Q., Boundary Element Analysis, (2009), Science Press Beijing |
| [45] | Hsiao, G. C.; Wendland, W. L., Boundary Integral Equations, (2008), Springer Berlin · Zbl 1157.65066 |
| [46] | Krysl, P.; Belytschko, T., Analysis of thin plates by the element-free Galerkin method, Comput. Mech., 17, 26-35, (1996) · Zbl 0841.73064 |
| [47] | 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 |
| [48] | Brenner, S. C.; Scott, L. R., The Mathematical Theory of Finite Element Methods, (1996), Springer-Verlag New York |
| [49] | Lions, J. L.; Magenes, E., Non-Homogeneous Boundary Value Problems and Applications, (1972), Springer Berlin · Zbl 0223.35039 |
| [50] | Zienkiewicz, O. C.; Taylor, R. L., The Finite Element Method, (2000), Butterworth Heinemann Oxford · Zbl 0991.74002 |
| [51] | Liu, G. R., Meshfree Methods: Moving beyond the Finite Element Method, (2009), CRC Press Boca Raton |
| [52] | Mathews, J. H.; Fink, K. K., Numerical Methods using MATLAB, (2002), Prentice Hall Englewood Cliffs |
| [53] | Li, X.; Zhu, J., The method of fundamental solutions for nonlinear elliptic problems, Eng. Anal. Bound. Elem. , 33, 322-329, (2009) · Zbl 1244.65221 |
| [54] | Zhu, T.; Zhang, J.; Atluri, S. N., A meshless local boundary integral equation (LBIE) method for solving nonlinear problems, Comput. Mech., 22, 174-186, (1998) · Zbl 0924.65105 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
