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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.

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
Full Text: DOI
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