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A localized approach for the method of approximate particular solutions. (English) Zbl 1221.65316
Summary: The method of approximate particular solutions (MAPS) has been recently developed to solve various types of partial differential equations. In the MAPS, radial basis functions play an important role in approximating the forcing term. Coupled with the concept of particular solutions and radial basis functions, a simple and effective numerical method for solving a large class of partial differential equations can be achieved. One of the difficulties of globally applying MAPS is that this method results in a large dense matrix which in turn severely restricts the number of interpolation points, thereby affecting our ability to solve large-scale science and engineering problems.In this paper we develop a localized scheme for the method of approximate particular solutions (LMAPS). The new localized approach allows the use of a small neighborhood of points to find the approximate solution of the given partial differential equation. In this paper, this local numerical scheme is used for solving large-scale problems, up to one million interpolation points. Three numerical examples in two-dimensions are used to validate the proposed numerical scheme.

##### MSC:
 65N99 Numerical methods for partial differential equations, boundary value problems 35J25 Boundary value problems for second-order elliptic equations
##### Keywords:
LMAPS; GMAPS; kd-tree; sparse system; particular solutions
Stony Brook
Full Text:
##### References:
 [1] Wen, P.H.; Chen, C.S., The method of particular solutions for solving scalar wave equations, The international journal for numerical methods in biomedical engineering, 26, 1878-1889, (2010) · Zbl 1208.65153 [2] Atkinson, K.E., The numerical evaluation of particular solutions for poisson’s equation, IMA journal of numerical analysis, 5, 319-338, (1985) · Zbl 0576.65114 [3] Chen, W., Symmetric boundary knot method, Engineering analysis with boundary elements, 26, 489-494, (2002) · Zbl 1006.65500 [4] Chen, W.; Hon, Y.C., Numerical convergence of boundary knot method in the analysis of Helmholtz, modified Helmholtz, and convection – diffusion problems, Computer methods in applied mechanics and engineering, 192, 1859-1875, (2003) · Zbl 1050.76040 [5] Golberg, M.A.; Chen, C.S., The method of fundamental solutions for potential, Helmholtz and diffusion problems, (), 103-176 · Zbl 0945.65130 [6] Chen, C.S.; Fan, C.M.; Wen, P.H., The method of particular solutions for solving certain partial differential equations, Numerical methods for partial differential equations, (2010) · Zbl 1242.65267 [7] Katsikadelis, John.T., The 2d elastostatic problem in inhomogeneous anisotropic bodies by the meshless analog equation method (maem), Engineering analysis with boundary elements: special issue: BEM/MRM for inhomogeneous solids, 32, 997-1005, (2008) · Zbl 1244.74220 [8] Mai-Duy, N.; Tran-Cong, T., Indirect RBFN method with thin plate splines for numerical solution of differential equations, Computer modeling in engineering & science, 4, 85-102, (2003) · Zbl 1148.76351 [9] Divo, E.; Kassab, A.J., An efficient localized RBF meshless method for fluid flow and conjugate hear transfer, ASME journal of heat transfer, 129, 124-136, (2007) [10] Lee, C.K.; Liu, X.; Fan, S.C., Local multiquadric approximation for solving boundary value problems, Computational mechanics, 30, 396-409, (2003) · Zbl 1035.65136 [11] Sarler, B.; Vertnik, R., Meshfree explicit local radial basis function collocation method for diffusion problems, Computers and mathematics with applications, 21, 1269-1282, (2006) · Zbl 1168.41003 [12] Shu, C.; Ding, H.; Yeo, K.S., Local radial basis function-based differential quadrature method and its application to solve two dimensional incompressible Navier-stokeequations, Computer methods and applied mechanics engineering, 192, 941-954, (2003) · Zbl 1025.76036 [13] Vertnik, R.; Sarler, B., Meshless local radial basis function collocation method for convective-diffusive solid – liquid phase change problems, International journal of numerical methods for heat and fluid flow, 16, 617-640, (2006) · Zbl 1121.80014 [14] Kansa, E.J.; Hon, Y.C., Circumventing the ill-conditioning problem with multiquadric radial basis functions: applications to elliptic partial differential equations, Computcomputer & mathematics. with applications, 39, 7-8, 123-137, (2000) · Zbl 0955.65086 [15] Hon, Y.C.; Schaback, R.; Zhou, X., An adaptive greedy algorithm for solving large RBF collocation problems, Numerical algorithms, 32, 1, 13-25, (2003) · Zbl 1019.65093 [16] Chen, C.S.; Golberg, M.A.; Ganesh, M.; Cheng, A.H.-D., Multilevel compact radial functions based computational schemes for some elliptic problems, Computers and mathematics with application, 43, 359-378, (2002) · Zbl 0999.65143 [17] Huang, C.-S.; Lee, Cheng-Feng; Cheng, A.H.-D., Error estimate, optimal shape factor, and high precision computation of multiquadric collocation method, Engineering analysis with boundary elements, 31, 7, 614-623, (2007) · Zbl 1195.65176 [18] Libre, N.A.; Emdadi, A.; Kansa, E.J.; Rahimian, M.; Shekarchi, M., A stabilized RBF collocation scheme for Neumann type boundary value problems, Computer modeling in engineering & science, 24, 61-80, (2008) · Zbl 1232.65156 [19] Cheng, A.H.-D.; Young, D.-L.; Tsai, J.-J., Solution of poisson’s equation by iterative DRBEM using compactly supported, positive definite radial basis function, Engineering analysis with boundary elements, 24, 549-557, (2000) · Zbl 0966.65089 [20] Beatson, R.K.; Greengard, L., A short course on fast multipole methods, (), 1-37 · Zbl 0882.65106 [21] Ling, Leevan; Schaback, Robert, An improved subspace selection algorithm for meshless collocation methods, International journal for numerical methods in engineering, 80, 1623-1639, (2009) · Zbl 1183.65153 [22] Rippa, Shmuel, An algorithm for selecting a good value for the parameter $$c$$ in radial basis function interpolation, Advances in computational mathematics, 11, 2-3, 193-210, (1999) · Zbl 0943.65017 [23] Wertz, J.; Kansa, E.J.; Ling, L., The role of the multiquadric shape parameters in solving elliptic partial differential equations, Computer & mathematical with applications, 51, 8, 1335-1348, (2006) · Zbl 1146.65078 [24] Karageorghis, A.; Chen, C.S.; Smyrlis, Y-S, Matrix decomposition RBF algorithm for solving 3d elliptic problems, Engineering analysis with boundary elements, 33, 1368-1373, (2009) · Zbl 1244.65184 [25] Duchon, J., Splines minimizing rotation invariant semi-norms in Sobolev spaces: constructive theory of functions of several variables, (), 85-110 [26] Skiena, Steven, The algorithm design manual, (2008), Springer · Zbl 1149.68081
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