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Applicability of the method of fundamental solutions. (English) Zbl 1244.65220
Summary: The condition number of a matrix is commonly used for investigating the stability of solutions to linear algebraic systems. Recent meshless techniques for solving partial differential equations have been known to give rise to ill-conditioned matrices, yet are still able to produce results that are close to machine accuracy. In this work, we consider the method of fundamental solutions (MFS), which is known to solve, with extremely high accuracy, certain partial differential equations, namely those for which a fundamental solution is known. To investigate the applicability of the MFS, either when the boundary is not analytic or when the boundary data are not harmonic, we examine the relationship between its accuracy and the effective condition number.
Three numerical examples are presented in which various boundary value problems for the Laplace equation are solved. We show that the effective condition number, which estimates system stability with the right-hand side vector taken into account, is roughly inversely proportional to the maximum error in the numerical approximation. Unlike the proven theories in literature, we focus on cases when the boundary and the data are not analytic. The effective condition number numerically provides an estimate of the quality of the MFS solution without any knowledge of the exact solution and allows the user to decide whether the MFS is, in fact, an appropriate method for a given problem, or what is the appropriate formulation of the given problem.

65N80 Fundamental solutions, Green’s function methods, etc. for boundary value problems involving PDEs
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[1] Kupradze, V.; Aleksidze, M., 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
[2] Golberg, M.A., The method of fundamental solutions for Poisson’s equation, Eng anal bound elem, 16, 3, 205-213, (1995)
[3] Balakrishnan, K.; Ramachandran, P.A., Osculatory interpolation in the method of fundamental solution for nonlinear Poisson problems, J comput phys, 172, 1, 1-18, (2001) · Zbl 0992.65131
[4] Chen, C.S., The method of fundamental solutions for non-linear thermal explosions, Commun numer methods eng, 11, 8, 675-681, (1995) · Zbl 0839.65143
[5] Partridge, P.W.; Sensale, B., The method of fundamental solutions with dual reciprocity for diffusion and diffusion – convection using subdomains, Eng anal bound elem, 24, 9, 633-641, (2000) · Zbl 1005.76064
[6] Kondapalli, P.S.; Shippy, D.J.; Fairweather, G., Analysis of acoustic scattering in fluids and solids by the method of fundamental solutions, J acoust soc am, 91, 4 Pt 1, 1844-1854, (1992)
[7] Kondapalli, P.S.; Shippy, D.J.; Fairweather, G., The method of fundamental solutions for transmission and scattering of elastic waves, Comput methods appl mech eng, 96, 2, 255-269, (1992) · Zbl 0825.73150
[8] Hon, Y.; Wei, T., A fundamental solution method for inverse heat conduction problem, Eng anal bound elem, 28, 5, 489-495, (2004) · Zbl 1073.80002
[9] Wei, T.; Hon, Y.C.; Ling, L., Method of fundamental solutions with regularization techniques for Cauchy problems of elliptic operators, Eng anal bound elem, 31, 2, 163-175, (2007)
[10] Fairweather, G.; Karageorghis, A., The method of fundamental solutions for elliptic boundary value problems, Adv comput math, 9, 1-2, 69-95, (1998) · Zbl 0922.65074
[11] Fairweather, G.; Karageorghis, A.; Martin, P., The method of fundamental solutions for scattering and radiation problems, Eng anal bound elem, 27, 7, 759-769, (2003) · Zbl 1060.76649
[12] Golberg, M.A.; Chen, C.S., The method of fundamental solutions for potential, Helmholtz and diffusion problems, (), 103-176 · Zbl 0945.65130
[13] Li, J.; Hon, Y.C.; Chen, C.S., Numerical comparisons of two meshless methods using radial basis functions, Eng anal bound elem, 26, 205-225, (2002) · Zbl 1003.65132
[14] Katsurada, M., A mathematical study of the charge simulation method. II, J fac sci univ Tokyo sect IA math, 36, 1, 135-162, (1989) · Zbl 0681.65081
[15] Katsurada, M.; Okamoto, H., A mathematical study of the charge simulation method. I, J fac sci univ Tokyo sect IA math, 35, 3, 507-518, (1988) · Zbl 0662.65100
[16] Kitagawa, T., On the numerical stability of the method of fundamental solution applied to the Dirichlet problem, Jpn J appl math, 5, 123-133, (1988) · Zbl 0644.65060
[17] Katsurada, M., Asymptotic error analysis of the charge simulation method in a Jordan region with an analytic boundary, J fac sci univ Tokyo sect IA math, 37, 3, 635-657, (1990) · Zbl 0723.65093
[18] Turing, A.M., Rounding-off errors in matrix processes, Quart J mech appl math, 1, 287-308, (1948) · Zbl 0033.28501
[19] Trefethen, L.N.; Bau, D.I., Numerical linear algebra, vol. xii, (1997), Society for Industrial and Applied Mathematics Philadelphia, PA, 361pp
[20] Christiansen, S., Condition number of matrices derived from two classes of integral equations, Math methods appl sci, 3, 364-392, (1981) · Zbl 0485.65089
[21] Chen, C.S.; Cho, H.A.; Golberg, M.A., Some comments on the ill-conditioning of the method of fundamental solutions, Eng anal bound elem, 30, 5, 405-410, (2006) · Zbl 1187.65136
[22] Chan, T.F.; Foulser, D.E., Effectively well-conditioned linear systems, SIAM J sci stat comput, 9, 6, 963-969, (1988) · Zbl 0664.65041
[23] Christiansen, S.; Hansen, P.C., The effective condition number applied to error analysis of certain boundary collocation methods, J comput appl math, 54, 1, 15-36, (1994) · Zbl 0834.65033
[24] Christiansen, S.; Saranen, J., The conditioning of some numerical methods for first kind boundary integral equations, J comput appl math, 67, 1, 43-58, (1996) · Zbl 0854.65104
[25] Banoczi, J.M.; Chiu, N.-C.; Cho, G.E.; Ipsen, I.C.F., The lack of influence of the right-hand side on the accuracy of linear system solution, SIAM J sci comput, 20, 1, 203-227, (1998) · Zbl 0914.65047
[26] Huang, H.T.; Li, Z.C., Effective condition number and superconvergence of the Trefftz method coupled with high order FEM for singularity problems, Eng anal bound elem, 30, 4, 270-283, (2006) · Zbl 1195.65151
[27] Li, Z.-C.; Chien, C.-S.; Huang, H.-T., Effective condition number for finite difference method, J comput appl math, 198, 1, 208-235, (2007) · Zbl 1104.65043
[28] Axler, S.; Bourdon, P.; Ramey, W., Harmonic function theory, (2000), Springer Berlin
[29] Schaback, R., Error estimates and condition numbers for radial basis function interpolation, Adv comput math, 3, 3, 251-264, (1995) · Zbl 0861.65007
[30] Schaback R. Adaptive numerical solution of MFS systems, 2007, preprint.
[31] Gaier D. Lectures on complex approximation, vol. XV [McLaughlin R, Translation from the German]. Boston, Basel, Stuttart: Birkhäuser; 1987. 196pp. · Zbl 0612.30003
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