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An equilibrated method of fundamental solutions to choose the best source points for the Laplace equation. (English) Zbl 1352.65637
Summary: For the method of fundamental solutions (MFS), a trial solution is expressed as a linear combination of fundamental solutions. However, the accuracy of MFS is heavily dependent on the distribution of source points. Two distributions of source points are frequently adopted: one on a circle with a radius $$R$$, and another along an offset $$D$$ to the boundary, where $$R$$ and $$D$$ are problem dependent constants. In the present paper, we propose a new method to choose the best source points, by using the MFS with multiple lengths $$R_k$$ for the distribution of source points, which are solved from an uncoupled system of nonlinear algebraic equations. Based on the concept of equilibrated matrix, the multiple-length $$R_k$$ is fully determined by the collocated points and a parameter $$R$$ or $$D$$, such that the condition number of the multiple-length MFS (MLMFS) can be reduced smaller than that of the original MFS. This new technique significantly improves the accuracy of the numerical solution in several orders than the MFS with the distribution of source points using $$R$$ or $$D$$. Some numerical tests for the Laplace equation confirm that the MLMFS has a good efficiency and accuracy, and the computational cost is rather cheap.

##### MSC:
 65N80 Fundamental solutions, Green’s function methods, etc. for boundary value problems involving PDEs
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##### References:
  Fairweather, G.; Karageorghis, A., The method of fundamental solutions for elliptic boundary value problems, Adv comput math, 9, 69-95, (1998) · Zbl 0922.65074  Saavedra, I.; Power, H., Multipole fast algorithm for the least-squares approach of the method of fundamental solutions for three-dimensional harmonic problems, Numer methods partial differential equation, 19, 828-845, (2003) · Zbl 1038.65127  Bogomolny, A., Fundamental solutions method for elliptic boundary value problems, SIAM J numer anal, 22, 644-669, (1985) · Zbl 0579.65121  Smyrlis, Y.S.; Karageorghis, A., Some aspects of the method of fundamental solutions for certain harmonic problems, J sci comput, 16, 341-371, (2001) · Zbl 0995.65116  Cho, H.A.; Golberg, M.A.; Muleshkov, A.S.; Li, X., Trefftz methods for time-dependent partial differential equations, CMC: comput mater contin, 1, 1-37, (2004)  Hon, Y.C.; Wei, T., The method of fundamental solution for solving multidimensional inverse heat conduction problems, CMES: comput model eng sci, 7, 119-132, (2005) · Zbl 1114.80004  Young, D.L.; Chen, K.H.; Lee, C.W., Novel meshless method for solving the potential problems with arbitrary domain, J comput phys, 209, 290-321, (2005) · Zbl 1073.65139  Young, D.L.; Ruan, J.W., Method of fundamental solutions for scattering problems of electromagnetic waves, CMES: comput model eng sci, 7, 223-232, (2005) · Zbl 1106.78008  Young, D.L.; Tsai, C.C.; Lin, Y.C.; Chen, C.S., The method of fundamental solutions for eigenfrequencies of plate vibrations, CMC: comput mater contin, 4, 1-10, (2006)  Chen CS, Karageorghis A, Smyrlis YS., editors. The method of fundamental solutions—a meshless method. Dynamic Publishers Inc.; 2008.  Liu, C.S., The method of fundamental solutions for solving the backward heat conduction problem with conditioning by a new post-conditioner, Numer heat transfer B: fundam, 60, 57-72, (2011)  Karageorghis, A.; Lesnic, D.; Marin, L., A survey of applications of the MFS to inverse problems, Inv prob sci eng, 19, 309-336, (2011) · Zbl 1220.65157  Golberg, M.A.; Chen, C.S., Discrete projection methods for integral equations, (1996), Computational Mechanics Publications Southampton  Chen, C.S.; Cho, H.A.; Golberg, M.A., Some comments on the ill-conditioning of the method of fundamental solutions, Eng anal boundary elem, 30, 405-410, (2006) · Zbl 1187.65136  Smyrlis, Y.S.; Karageorghis, A., Numerical analysis of the MFS for certain harmonic problems, M2AN math model numer anal, 38, 495-517, (2004) · Zbl 1079.65108  Alves, C.J.S.; Leitão, V.M.A., Crack analysis using an enriched MFS domain decomposition technique, Eng anal boundary elem, 30, 160-166, (2006) · Zbl 1195.74148  Tsai, C.C.; Lin, Y.C.; Young, D.L.; Atluri, S.N., Investigations on the accuracy and condition number for the method of fundamental solutions, CMES: comput model eng sci, 16, 103-114, (2006)  Young, D.L.; Chen, K.H.; Chen, J.T.; Kao, J.H., A modified method of fundamental solutions with source on the boundary for solving Laplace equations with circular and arbitrary domains, CMES: comput model eng sci, 19, 197-222, (2007) · Zbl 1184.65116  Ramachandran, P.A., Method of fundamental solutions: singular value decomposition analysis, Commun numer methods eng, 18, 789-801, (2002) · Zbl 1016.65095  Liu, C.S., Improving the ill-conditioning of the method of fundamental solutions for 2D Laplace equation, CMES: comput model eng sci, 28, 77-93, (2008) · Zbl 1232.65170  Liu, C.S., A modified Trefftz method for two-dimensional Laplace equation considering the Domain’s characteristic length, CMES: comput model eng sci, 21, 53-65, (2007) · Zbl 1232.65157  Liu, C.S., An effectively modified direct Trefftz method for 2D potential problems considering the Domain’s characteristic length, Eng anal boundary elem, 31, 983-993, (2007) · Zbl 1259.65183  Liu, C.S., A highly accurate collocation Trefftz method for solving the Laplace equation in the doubly connected domains, Numer methods partial differential equation, 24, 179-192, (2008) · Zbl 1130.65114  Shigeta, T.; Young, D.L., Mathematical and numerical studies on meshless methods for exterior unbounded domain problems, J comput phys, 230, 6900-6915, (2011) · Zbl 1252.65202  Christiansen, S., Condition number of matrices derived from two classes of integral equations, Math methods appl sci, 3, 364-392, (1981) · Zbl 0485.65089  Katsurada, M., A mathematical study of the charge simulation method II, J fac sci univ Tokyo sec 1A, 36, 135-162, (1989) · Zbl 0681.65081  Drombosky, T.W.; Meyer, A.L.; Ling, L., Applicability of the method of fundamental solutions, Eng anal boundary elem, 33, 637-643, (2009) · Zbl 1244.65220  Mathon, R.; Johnston, R.L., The approximate solution of elliptic boundary value problems by fundamental solutions, SIAM J numer anal, 14, 638-650, (1977) · Zbl 0368.65058  Johnston, R.L.; Fairweather, G., The method of fundamental solutions for problems in potential flow, Appl math model, 8, 265-270, (1984) · Zbl 0546.76021  Wang, J.G.; Ahmed, M.T.; Leavers, J.D., Nonlinear least squares optimization applied to the method of fundamental solutions for eddy current problems, IEEE trans magnet, 26, 2385-2387, (1990)  Katsurada, M.; Okamoto, H., The collocation points of the fundamental solution method for the potential problem, Comput math appl, 31, 123-137, (1996) · Zbl 0852.65101  Alves, C.J.S., On the choice of source points in the method of fundamental solutions, Eng anal boundary elem, 33, 1348-1361, (2009) · Zbl 1244.65216  Ling, L.; Opfer, R.; Schaback, R., Results on meshless collocation techniques, Eng anal boundary elem, 30, 247-253, (2006) · Zbl 1195.65177  Schaback, R., Adaptive numerical solution of MFS systems, (), 1-27  Kupradze, V.D.; Aleksidze, M.A., The method of functional equations for the approximate solution of certain boundary value problems, USSR comput math math phys, 4, 82-126, (1964) · Zbl 0154.17604  Liu, C.S.; Atluri, S.N., An iterative method using an optimal descent vector, for solving an ill-conditioned system bx={\bfb}, better and faster than the conjugate gradient method, CMES: comput model eng sci, 80, 275-298, (2011)  Bauer, F.L., Optimally scaled matrices, Numer math, 5, 73-87, (1963) · Zbl 0107.10501  van der Sluis, A., Condition numbers and equilibration of matrices, Numer math, 14, 14-23, (1969) · Zbl 0182.48906  Gautsch, W., Optimally scaled and optimally conditioned Vandermonde and Vandermonde-like matrices, BIT numer math, 51, 103-125, (2011) · Zbl 1214.65021  Liu, C.S.; Hong, H.K.; Atluri, S.N., Novel algorithms based on the conjugate gradient method for inverting ill-conditioned matrices, and a new regularization method to solve ill-posed linear systems, CMES: comput model eng sci, 60, 279-308, (2010) · Zbl 1231.65071  Motz, H., The treatment of singularities in relaxation methods, Q appl math, 4, 371-377, (1946) · Zbl 0030.40002  Li, Z.C., Combined methods for elliptic equations with singularities, interfaces and infinities, (1998), Kluwer Academic Publishers Netherlands · Zbl 0909.65079  Liu, C.S., A highly accurate solver for the mixed-boundary potential problem and singular problem in arbitrary plane domain, CMES: comput model eng sci, 20, 111-122, (2007)  Li, Z.C., The method of fundamental solutions for the Laplace equation with mixed boundary problems, (), 29-50
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