On condition number of meshless collocation method using radial basis functions. (English) Zbl 1090.65132

The following linear elliptic problem \(\mathcal L u=f\) in \(\Omega\), \(\mathcal B u=g\) on \(\partial\Omega\), where \(\Omega\) is a \(d\)-dimensional domain, with a smooth boundary \(\partial\Omega\), whose outer unit normal direction is denoted by \(\mathbf n\), \(f\) is a given function of \(L^2(\Omega)\), \(\mathcal L=\sum_{i,j}\partial_i(a_{i,j}\partial_j)+ \sum_i b_i\partial_i + C_0,\quad \mathcal B=\text{Id}\), or \({\partial\over \partial n_L}+ \text{Id}\), is considered. The authors give an estimate for the condition number of meshless collocation method using radial basis functions.


65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
65F35 Numerical computation of matrix norms, conditioning, scaling
35J25 Boundary value problems for second-order elliptic equations
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[1] Adams, R.A., Sobolev spaces, (1975), Academic Press · Zbl 0186.19101
[2] Fasshauer, G.E., Solving differential equations with radial basis functions: multilevel methods and smoothing, Adv. comput. math., 11, 139-159, (1999) · Zbl 0940.65122
[3] Fedoseyev, A.I.; Friedman, M.J.; Kansa, E.J., Improved multiquadric method for elliptic partial differential equations via PDE collocation on the boundary. radial basis functions and partial differential equations, Comput. math. appl., 43, 3-5, 439-455, (2002) · Zbl 0999.65137
[4] Franke, C.; Schaback, R., Convergence order estimates of meshless collocation methods using radial basis functions, Adv. comput. math., 8, 381-399, (1998) · Zbl 0909.65088
[5] Grisvard, P., Elliptic problems in nonsmooth domains, (1985), Pitman · Zbl 0695.35060
[6] Landau, H., Necessary density conditions for sampling and interpolation of certain entire functions, Acta math., 117, 37-52, (1967) · Zbl 0154.15301
[7] Schaback, R., Error estimates and condition numbers for radial basis function interpolation, Adv. comput. math., 3, 251-264, (1995) · Zbl 0861.65007
[8] Schaback, R., Improved error bounds for scattered data interpolation by radial basis functions, Math. comput., 68, 201-216, (1999) · Zbl 0917.41011
[9] Wendland, H., Sobolev-type error estimates for interpolation by radial basis functions, (), 337-344 · Zbl 0955.41002
[10] Wendland, H., Error estimates for interpolation by compactly supported radial basis functions of minimal degree, J. approx. theory, 93, 258-272, (1998) · Zbl 0904.41013
[11] Wendland, H., Meshless Galerkin methods using radial basis functions, Math. comput., 68, 1521-1531, (1999) · Zbl 1020.65084
[12] Duan, Yong; Tan, Yong-ji, Meshless Galerkin method based on regions partitioned into subdomains using global RBF, () · Zbl 1063.65122
[13] Yong Duan, Yong-ji Tan, Meshless Galerkin method for Dirichlet problems using radial basis function, preprint. · Zbl 1106.65103
[14] Hon, Y.C.; Wu, Z., Additive Schwarz domain decomposition with radial basis approximation, Int. J. appl. math., 4, 1, 81-98, (2000) · Zbl 1051.65121
[15] Hon, Y.C.; Schaback, R., On unsymmetric collocation by radial basis functions, Appl. math. comput., 119, 177-186, (2001) · Zbl 1026.65107
[16] Wu, Z., Compactly supported positive definite radial functions, Adv. comput. math., 4, 283-292, (1995) · Zbl 0837.41016
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