×

Application of the multiquadric method for numerical solution of elliptic partial differential equations. (English) Zbl 0883.65083

This paper uses the multiquadric (MQ) approximation scheme for the solution of elliptic partial differential equations with Dirichlet and/or Neumann boundary conditions. The scheme has the advantage of using the data points in arbitrary locations with an arbitrary ordering. Two-dimensional Laplace, Poisson, and biharmonic equations describing the various physical processes have been taken as the test examples. The agreement is found to be very good between the computed and exact solutions. The method also provides an excellent approximation with a curved boundary.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35J40 Boundary value problems for higher-order elliptic equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Nayfeh, A. H., Perturbation Methods (1973), Wiley Interscience: Wiley Interscience New York · Zbl 0375.35005
[2] Van Dyke, M., Perturbation Methods in Fluid Mechanics (1975), Parabolic Press: Parabolic Press Palo Alto · Zbl 0329.76002
[3] Kevorkian, J.; Cole, J. D., Perturbation methods in Applied Mathematics (1981), Springer-Verlag: Springer-Verlag New York · Zbl 0456.34001
[4] Parlange, J. Y., Theory of water movements in soils I: One-dimensional absorption, Soil Sci., 111, 134-170 (1971)
[5] Tsang, T., Transient state heat transfer and diffusion problems, Ind. Eng. Chem., 52, 707 (1960)
[6] Hardy, R. L., Theory and applications of the multiquadric biharmonic method: 20 Years of discovery 1968-1988, Comput. Math. Appl., 19, 8/9, 163-208 (1990) · Zbl 0692.65003
[7] Franke, R., Scattered data interpolation: tests of some methods, Math. Comp., 38, 181-200 (1982) · Zbl 0476.65005
[8] Kansa, E. J., Multiquadrics: a scattered data approximation scheme with application to computational fluid dynamics: 1. surface approximations and partial derivative estimates, Comput. Math. Applic., 19, 8/9, 127-154 (1990) · Zbl 0692.76003
[9] Kansa, E. J., Multiquadrics: a scattered data approximation scheme with application to computational fluid dynamics: II parabolic, hyperbolic, and elliptic partial differential equations, Comput. Math. Applic., 19, 8/9, 146-161 (1990) · Zbl 0850.76048
[10] Micchelli, C. A., Interpolation of scattered data: distance matrices and conditionally positive definite functions, Const. Approx., 2, 11-22 (1986) · Zbl 0625.41005
[11] Baxter, B. C., The asymptotic cardinal function of the multiquadric, \(f(r) = (r^2 + c^2)\) as \(c^2Æ ·\), Comput. Math. Applic., 24, 12, 1-6 (1992) · Zbl 0764.41016
[12] Madych, W. R., Miscellaneous error bounds for multiquadric and related interpolators, Comput. Math. Applic., 24, 12, 121-138 (1992) · Zbl 0766.41003
[13] Madych, W. R.; Nelson, S. A., Multivariate interpolation and conditionally positive definite functions, Approx. Theo. Appl., 4, 77-89 (1989) · Zbl 0703.41008
[14] Madych, W. R.; Nelson, S. A., Multivariate interpolation and conditionally positive definite functions II, Math. Comput., 54, 211-230 (1990) · Zbl 0859.41004
[15] Dubal, M. R., Construction of three-dimensional black-hole initial data via multiquadrics, Phys. Rev. D, 45, 117-118 (1992)
[16] Dubal, M. R.; Oliverirá, S. R.; Matzner, R. A., Solution of elliptic equations in numerical relativity using multiquadrics, (Inverno, R. D., Approaches to Numerical Relativity (1992), Cambridge University Press: Cambridge University Press Cambridge, England)
[17] Moridis, G.; Kansa, E. J., The Laplace transform multiquadric method highly accurate scheme for numerical solution of partial differential equations, J. Appl. Sci. and Comput., 1, 2, 375-407 (1994)
[18] Carlson, R. E.; Foley, T. A., The parameter \(R^2\) in multiquadric interpolation, Comput. Math. Applic., 21, 29-42 (1991) · Zbl 0725.65009
[19] Hagen, R. E.; Kansa, E. J., Studies of the \(R\) parameter in the multiquadric function applied to ground water pumping, J. Appl. Sci. and Comput., 1, 2, 266-281 (1994)
[20] Marquadt, D. M., An algorithm for least squares estimation of nonlinear parameters, J. Soc. Ind. Appl. Math., 11, 431-441 (1963) · Zbl 0112.10505
[21] M. A. Golberg, C. S. Chen, and S. R. Karur, Improved multiquadric approximation for partial differential equations, preprint; M. A. Golberg, C. S. Chen, and S. R. Karur, Improved multiquadric approximation for partial differential equations, preprint
[22] Makroglou, A., Radial basis functions in the numerical solution of Fredholm integral and intergrodifferential equations, (Vichnevetsky, R.; Knight, D.; Richter, G., Proc. of the 7th IMACS conference on Computer Methods for PDEs, New Brunswick, NJ, USA. Proc. of the 7th IMACS conference on Computer Methods for PDEs, New Brunswick, NJ, USA, Advances in Computer Methods for Partial Differential equations, VII (June 22-24 1992)), 478-484
[23] Carslaw, H. S.; Jaeger, J. C., Heat Conduction in Solids (1976), Oxford University Press: Oxford University Press Oxford · Zbl 0029.37801
[24] Hughes, W. F.; Gaylord, E. W., Basic equations of Engineering Sciences (1964), Schaum Publishing Company: Schaum Publishing Company New York
[25] Sharan, Maithili; Singh, M. P.; Sud, I., Modelling of oxygen transport in systemic circulation in a hyperbaric environment, Frontiers Med. Biol. Eng., 3, 2-3′ (1991)
[26] Press, W. H.; Flannery, B. P.; Teukolsy, S. A.; Vetterling, W. T., Numerical Recipes (1986), Cambridge University Press: Cambridge University Press Cambridge
[27] Foley, T. A., The map and blend scattered data interpolation on a sphere, Comput. Math. Applic., 24, 12, 41-60 (1992) · Zbl 0800.68712
[28] Dubal, M. R., Domain decomposition and local refinement for multiquadric approximations. I: Second-order equations in one dimension, J. Appl. Sci. and Comput., 1, 1, 146-171 (1994)
[29] Carlson, R. E.; Natarajan, B. K., Sparse approximate multiquadric interpolation, Comput. Math. Applic., 27, 6, 99-108 (1994) · Zbl 0801.65008
[30] Franke, R.; Hagen, H.; Nielson, G. M., Least squares surface approximation in scattered data using multiquadric function, Adv. Comput. Math., 2, 81-99 (1994) · Zbl 0831.65015
[31] Franke, R.; Hagen, H.; Nielson, G. M., Repeated knots in least squares multiquadric functions, Computing Suppl., 10 (1995) · Zbl 0839.65010
[32] Kansa, E. J.; Carlson, R. E., Improved accuracy of multiquadric interpolation using variable shape parameters, Comput. Math. Applic., 24, 12, 99-120 (1992) · Zbl 0765.65008
[33] Girosi, F., Some extensions of radial basis functions and the applications to artificial intelligence, Comput. Math. Applic., 24, 12, 61-80 (1992) · Zbl 0800.68670
[34] Buhmann, M. D., Discrete least squares approximation and prewavelets for radial function spaces, (DAMTP 1993 / NA4 (1993), University of Cambridge: University of Cambridge Cambridge, England) · Zbl 0790.41015
[35] Galperin, E. A.; Zheng, Q., Solution and control of PDE via Global optimization methods, Comput. Math. Applic., 25, 5, 103-111 (1993) · Zbl 0794.35009
[36] Galperin, E. A., The cubic algorithm for optimization and control (1990), ND Researchè Publ: ND Researchè Publ Montreal · Zbl 0781.90080
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.