×

zbMATH — the first resource for mathematics

The role of the multiquadric shape parameters in solving elliptic partial differential equations. (English) Zbl 1146.65078
Summary: This study examines the generalized multiquadrics (MQ), \(\varphi_j(x) = [(x-x_j)^2+c_j^2]^\beta\) in the numerical solutions of elliptic two-dimensional partial differential equations (PDEs) with Dirichlet boundary conditions. The exponent \(\beta\) as well as \(c_j^2\) can be classified as shape parameters since these affect the shape of the MQ basis function. We examine variations of \(\beta\) as well as \(c_j^2\) where \(c_j^2\) can be different over the interior and on the boundary. The results show that increasing \(\beta\) has the most important effect on convergence, followed next by distinct sets of \((c_j^2)_{\Omega\setminus\partial\Omega}\ll(c_j^2)_{\partial\Omega}\). Additional convergence accelerations are obtained by permitting both \((c_j^2)_{\Omega\setminus\partial\Omega}\) and \((c_j^2)_{\partial\Omega}\) to oscillate about its mean value with amplitude of approximately 1/2 for odd and even values of the indices. Our results show high orders of accuracy as the number of data centers increases with some simple heuristics.

MSC:
65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Fedoseyev, A.I.; Friedman, M.J.; Kansa, E.J., Improved multiquadric method for elliptic partial differential equations via PDE collocation on the boundary, Computers math. applic., 43, 3-5, 491-500, (2002) · Zbl 0999.65137
[2] Cheng, A.H.D.; Golberg, M.A.; Kansa, E.J.; Zammito, T., Exponential convergence and h-c multiquadric collocation method for partial differential equations, Num. meth. pdes, 19, 571-594, (2003) · Zbl 1031.65121
[3] Wu, Y.L.; Shu, C., Development of RBF-DQ method for derivative approximation and its application to simulate natural convection in concentric annuli, Comput. mech., 29, 477-485, (2002) · Zbl 1146.76635
[4] Shu, C.; Ding, H.; Yee, K.S., Local radial basis function based differential quadrature method and its applications to solve two-dimensional incompressible Navier-Stokes equations, Comput. meth. appl. mech. engng., 192, 941-954, (2003) · Zbl 1025.76036
[5] Madych, W.R.; Nelson, S.A., Multivariate interpolation and conditionally positive definite functions, ap-prox, Theory appl., 4, 77-89, (1988) · Zbl 0703.41008
[6] Madych, W.R., Miscellaneous error bounds for multiquadric and related interpolators, Computers math. applic., 24, 12, 121-138, (1992) · Zbl 0766.41003
[7] Kansa, E.J., Multiquadrics-A scattered data approximation scheme with applications to computational fluid dynamics: H. solutions to parabolic, hyperbolic, and elliptic partial differential equations, Computers math. applic., 19, 6-8, 147-161, (1990) · Zbl 0850.76048
[8] Golberg, M.A.; Chen, C.S.; Karur, S.R., Improved multiquadric approximation for partial differential equations, Eng. anal. bound. elem., 18, 9-17, (1996)
[9] Hardy, R.L., Multiquadric equations of topography and other irregular surfaces, J. geophys. res., 76, 1905-1915, (1971)
[10] Foley, T.A., Interpolation and approximation of 3-D and 4-D scattered data, Computers math. applic., 13, 8, 711-740, (1987) · Zbl 0635.65007
[11] Carlson, R.E.; Foley, T.A., The parameter R2 in multiquadric interpolation, Computers math. applic., 21, 9, 29-42, (1991) · Zbl 0725.65009
[12] Kansa, E.J.; Carlson, R.E., Improved accuracy of multiquadric interpolation using variable shape parameters, Computers math. applic., 24, 12, 99-120, (1992) · Zbl 0765.65008
[13] Rippa, S., An algorithm for selecting a good value for the parameter c in radial basis functions interpolation, Adv. comp. math., 11, 193-210, (1999) · Zbl 0943.65017
[14] Kansa, E.J.; Hon, Y.C., Circumventing the ill-conditioning problem with multiquadric radial basis functions: applications to elliptic partial differential equations, Computers math. applic., 39, 7/8, 123-137, (2000) · Zbl 0955.65086
[15] Driscoll, T.A.; Fornberg, B., Interpolation in the limit of increasingly.at radial basis functions, Computers math. applic., 43, 3-5, 413-422, (2002) · Zbl 1006.65013
[16] Fornberg, B.; Wright, G.; Larsson, E., A numerical study of some radial basis function based solution methods for elliptic pdes, Computers math. applic., 46, 5/6, 891-902, (2003) · Zbl 1049.65136
[17] Fornberg, B.; Wright, G.; Larsson, E., Some observations regarding interpolants in the limit of at radial basis functions, Computers math. applic., 47, 1, 37-55, (2004) · Zbl 1048.41017
[18] Larsson, E.; Fornberg, B., Theoretical and computational aspects of multivariate interpolation with increasingly at radial basis functions, Computers math. applic., 49, 1, 103-130, (2005) · Zbl 1074.41012
[19] Wang, J.G.; Liu, G.R., On the optimal shape parameters of radial basis functions used for 2-D meshless methods, Comput. meth. appl. mech. eng., 191, 2611-2630, (2002) · Zbl 1065.74074
[20] Xiao, J.R.; McCarthy, M.A., A local heaviside weighted meshless method for two-dimensional solids using radial basis functions, Computat. mech., 31, 301-315, (2003) · Zbl 1038.74680
[21] Xiao, J.R.; Gama, B.A.; Gillespie, J.W.; Kansa, E.J., Meshless solutions of 2D contact problems by subdomain variational inequality and MLPG method with radial basis functions, Eng. anal. bound. elem., 29, 95-106, (2005) · Zbl 1182.74261
[22] Ling, L.; Kansa, E.J., A least squares preconditioner for radial basis functions collocation methods, Adv. comput. math., 23, 31-54, (2005) · Zbl 1067.65136
[23] Ling, L.; Kansa, E.J., Preconditioning for radial basis functions with domain decomposition, Mathl. comput. modelling, 40, 13, 1413-1427, (2004) · Zbl 1077.41008
[24] Brown, D.; Ling, L.; Kansa, E.J.; Levesley, J., On approximate cardinal preconditioning methods for radial functions, Eng. anal. bound. elem., 29, 343-353, (2005) · Zbl 1182.65174
[25] Ling, L.; Hon, Y.C., Improved numerical solver for Kansa’s method based on affine space decomposition, Eng. anal. bound. elem., 29, 1077-1085, (2005) · Zbl 1182.65176
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.