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Convergent Overdetermined-RBF-MLPG for Solving Second Order Elliptic PDEs

Published online by Cambridge University Press:  03 June 2015

Ahmad Shirzadi*
Affiliation:
Department of Mathematics, Persian Gulf University, Bushehr, Iran
Leevan Ling*
Affiliation:
Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong
*
Corresponding author. Email: shirzadi.a@gmail.com
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Abstract

This paper deals with the solvability and the convergence of a class of unsymmetric Meshless Local Petrov-Galerkin (MLPG) method with radial basis function (RBF) kernels generated trial spaces. Local weak-form testings are done with step-functions. It is proved that subject to sufficiently many appropriate testings, solvability of the unsymmetric RBF-MLPG resultant systems can be guaranteed. Moreover, an error analysis shows that this numerical approximation converges at the same rate as found in RBF interpolation. Numerical results (in double precision) give good agreement with the provided theory.

Type
Research Article
Copyright
Copyright © Global-Science Press 2013

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References

[1]Atluri, S. and Zhu, T., A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics, Comput. Mech., 22(2) (1998), pp. 117127.CrossRefGoogle Scholar
[2]Atluri, S. and Zhu, T., A new meshless local Petrov-Galerkin (MLPG) approach to nonlinear problems in computer modeling and simulation, Comput. Model. Sim. Eng., 3 (1998), pp. 187196.Google Scholar
[3]Dehghan, M. and Mirzaei, D., Meshless local Petrov-Galerkin (MLPG) method for the unsteady magnetohydrodynamic (MHD) flow through pipe with arbitrary wall conductivity, Appl. Numer. Math., 59(5) (2009), pp. 10431058.CrossRefGoogle Scholar
[4]Mirzaei, D. and Dehghan, M., A meshless based method for solution of integral equations, Appl. Numer. Math., 60(3) (2010), pp. 245262.Google Scholar
[5]Sladek, J., Sladek, V. and Hon, Y., Inverse heat conduction problems by meshless local Petrov-Galerkin method, Eng. Anal. Bound. Elem., 30(8) (2006), pp. 650661.Google Scholar
[6]Abbasbandy, S., Sladek, V., Shirzadi, A. and Sladek, J., Numerical simulations for coupled pair of diffusion equations by mlpg method, CMES Compt. Model. Eng. Sci., 71(1) (2011), pp. 1537.Google Scholar
[7]Schaback, R., Unsymmetric meshless methods for operator equations, Numer. Math., 114 (2010), pp. 629651.Google Scholar
[8]Atluri, S., Kim, H.-G. and Cho, J. Y., A critical assessment of the truly meshless local Petrov-Galerkin (MLPG) and local boundary integral equation (lbie) methods, Comput. Mech., 24 (1999), pp. 348372.Google Scholar
[9]Abbasbandy, S. and Shirzadi, A., A meshless method for two-dimensional diffusion equation with an integral condition, Eng. Anal. Bound. Elem., 34(12) (2010), pp. 10311037.Google Scholar
[10]Abbasbandy, S. and Shirzadi, A., MLPG method for two-dimensional diffusion equation with Neumann’s and non-classical boundary conditions, Appl. Numer. Math., 61 (2011), pp. 170180.Google Scholar
[11]Gu, Y., Chen, W. and Zhang, C.-Z., Sing./ular boundary method for solving plane strain elasto-static problems, Int. J. Solids Struct., 48(18) (2011), pp. 25492556.Google Scholar
[12]Chen, W. and Tanaka, M., A meshless, integration-free, and boundary-only rbf technique, Comput. Math. Appl., 43 (2002), pp. 379391.Google Scholar
[13]Chen, W., New rbf collocation schemes and kernel rbfs with applications, in: Lecture Notes in Computational Science and Engineering, (2002), pp. 7586.Google Scholar
[14]Raju, I. S., Phillips, D. R. and Krishnamurthy, T., A radial basis function approach in the meshless local Petrov-Galerkin method for euler-bernoulli beam problems, Comput. Mech., 34 (2004), pp. 464474.Google Scholar
[15]Atluri, S. N. and Shen, S., The basis of meshless domain discretization: the meshlessPetrov-Galerkin (MLPG) method, Adv. Comput. Math., 23 (2005), pp. 7393.Google Scholar
[16]Ling, L., Opfer, R. and Schaback, R., Results on meshless collocation techniques, Eng. Anal. Bound. Elem., 30(4) (2006), pp. 247253.Google Scholar
[17]Schaback, R., Error estimates and condition numbers for radial basis function interpolation, Adv. Comput. Math., 3(3) (1995), pp. 251264.Google Scholar
[18]Schaback, R., Approximation by radial basis functions with finitely many centers, Construct. Approximation, 12(3) (1996), pp. 331340.Google Scholar
[19]Madych, W. R. and Nelson, S. A., Multivariate interpolation and conditionally positive definite functions, Approximation Theory Appl., 4(4) (1988), pp. 7789.Google Scholar
[20]Gilbarg, D. and Trudinger, N. S., Elliptic Partial Differential Equations of Second Order, Springer, 1998.Google Scholar
[21]Ling, L. and Schaback, R., Stable and convergent unsymmetric meshless collocation methods, SIAM J. Numer. Anal., 46(3) (2008), pp. 10971115.Google Scholar
[22]Lee, C. F., Ling, L. and Schaback, R., On convergent numerical algorithms for unsymmetric collocation, Adv. Comput. Math., 30(4) (2009), pp. 339354.Google Scholar
[23]Huang, C.-S., Lee, C.-F. and Cheng, A.-D., Error estimate, optimal shape factor and high precision computation of multiquadric collocation method, Eng. Anal. Bound. Elem., 31(7) (2007), pp. 614623.Google Scholar
[24]Ling, L. and Schaback, R., An improved subspace selection algorithm for meshless collocation methods, Int. J. Numer. Methods Eng., 80(13) (2009), pp. 16231639.Google Scholar
[25]Kwok, T. O. and Ling, L., On convergence of a least-squares Kansa’s method for the modified Helmholtz equations, Adv. Appl. Math. Mech., 1(3) (2009), pp. 367382.Google Scholar