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Error estimate, optimal shape factor, and high precision computation of multiquadric collocation method. (English) Zbl 1195.65176
Summary: Multiquadric (MQ) collocation method is highly efficient for solving partial differential equations due to its exponential error convergence rate. A special feature of the method is that error can be reduced by increasing the value of shape constant \(c\) in the MQ basis function, without refining the grid. It is believed that in a numerical solution without roundoff error, infinite accuracy can be achieved by letting \(c\rightarrow \infty \). Using the arbitrary precision computation, this paper tests the above conjecture. A sharper error estimate than previously obtained is presented. A formula for a finite, optimal \(c\) value that minimizes the solution error for a given grid size is obtained. Using residual errors, constants in error estimate and optimal \(c\) formula can be obtained. These results are supported by numerical examples.

65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
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[1] Kansa, E.J., Multiquadrics—a scattered data approximation scheme with applications to computational fluid-dynamics, I, Comput math appl, 19, 127-145, (1990) · Zbl 0692.76003
[2] Cheng, A.H.D.; Golberg, M.A.; Kansa, E.J.; Zammito, G., Exponential convergence and h-c multiquadric collocation method for partial differential equations, Numer methods partial differential equations, 19, 5, 571-594, (2003) · Zbl 1031.65121
[3] Cheng, A.H.D.; Cabral, J.J.S.P., Direct solution of ill-posed boundary value problems by radial basis function collocation method, Int J numer methods eng, 64, 1, 45-64, (2005) · Zbl 1108.65059
[4] Boyd, J.P., Chebyshev and Fourier spectral methods, (2001), Dover New York · Zbl 0994.65128
[5] Bogomolny, A., Fundamental solutions method for elliptic boundary value problems, SIAM J numer anal, 22, 644-669, (1985) · Zbl 0579.65121
[6] Madych, W.R., Miscellaneous error bounds for multiquadric and related interpolators, Comput math appl, 24, 121-138, (1992) · Zbl 0766.41003
[7] Brown, D.; Ling, L.; Kansa, E.J.; Levesley, J., On approximate cardinal preconditioning methods for solving PDEs with radial basis functions, Eng anal bound elem, 29, 4, 343-353, (2005) · Zbl 1182.65174
[8] Fornberg, B.; Wright, G.; Larsson, E., Some observations regarding interpolants in the limit of flat radial basis functions, Comput math appl, 47, 1, 37-55, (2004) · Zbl 1048.41017
[9] Fornberg, B.; Wright, G., Stable computation of multiquadric interpolants for all values of the shape parameter, Comput math appl, 48, 5-6, 853-867, (2004) · Zbl 1072.41001
[10] Kansa, E.J.; Carlson, R.E., Improved accuracy of multiquadric interpolation using variable shape parameters, Comput math appl, 24, 99-120, (1992) · Zbl 0765.65008
[11] Kansa, E.J.; Hon, Y.C., Circumventing the ill-conditioning problem with multiquadric radial basis functions: applications to elliptic partial differential equations, Comput math appl, 39, 123-137, (2000) · Zbl 0955.65086
[12] Larsson, E.; Fornberg, B., Theoretical and computational aspects of multivariate interpolation with increasingly flat radial basis functions, Comput math appl, 49, 1, 103-130, (2005) · Zbl 1074.41012
[13] Ling, L.; Hon, Y.C., Improved numerical solver for Kansa’s method based on affine space decomposition, Eng anal boun elem, 29, 12, 1077-1085, (2005) · Zbl 1182.65176
[14] Hsiao, G.C.; Kleinman, R.E.; Li, R.X.; van den Burg, P.M., Residual error—a simple and sufficient estimate of actual error in solutions of boundary integral equations, (), 73-83
[15] Wilkinson, J.H., The algebraic eigenvalue problems, (1965), Oxford University Press Oxford · Zbl 0258.65037
[16] Atkinson, K.E., An introduction to numerical analysis, (1989), Wiley New York · Zbl 0718.65001
[17] Chan, T.F.; Fousler, D.E., Effectively well-conditioned linear system, SIAM J sci stat comput, 9, 6, 963-969, (1988) · Zbl 0664.65041
[18] 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
[19] Li ZC, Lu ZZ, Hu HY, Cheng AHD. Trefftz and collocation method. Southampton: WIT Press, in press.
[20] Bailey DH, Hida Y, Jeyabalan K, Li XS, Thompson B. High precision software directory, \(\langle\)http://crd.lbl.gov/\(\sim\)dhbailey/mpdist/⟩.
[21] Golub, G.H.; van Loan, C.F., Matrix computations, (1989), Johns Hopkins University Press Baltimore · Zbl 0733.65016
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