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Exponential convergence and \(h\)-\(c\) multiquadric collocation method for partial differential equations. (English) Zbl 1031.65121
Summary: The radial basis function (RBF) collocation method uses global shape functions to interpolate and collocate the approximate solution of partial differential equations. It is a truly meshless method as compared to some of the so-called meshless or element-free finite element methods. For the multiquadric and Gaussian RBFs, there are two ways to make the solution converge – either by refining the mesh size \(h\), or by increasing the shape parameter \(c\). While the \(h\)-scheme requires the increase of computational cost, the \(c\)-scheme is performed without extra effort.
In this paper we establish by numerical experiment the exponential error estimate \(\varepsilon\sim O(\lambda^{\sqrt{c/h}})\), where \(0<\lambda< 1\). We also propose the use of residual error as an error indicator to optimize the selection of \(c\).

MSC:
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
65N15 Error bounds for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
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