Applied Mathematics and Computation
Solving partial differential equations by collocation using radial basis functions
Applied Mathematics and Computation
Meshless Galerkin methods using radial basis functions
Mathematics of Computation
Combinations of collocation and finite-element methods for Poisson's equation
Computers & Mathematics with Applications
Coupling three-field formulation and meshless mixed Galerkin methods using radial basis functions
Journal of Computational and Applied Mathematics
Scattered data interpolation and approximation for computer graphics
ACM SIGGRAPH ASIA 2010 Courses
Journal of Computational Physics
Computers & Mathematics with Applications
Computers & Mathematics with Applications
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This paper introduces radial basis functions (RBF) into the collocation methods and the combined methods for elliptic boundary value problems. First, the Ritz-Galerkin method (RGM) is chosen using the RBF, and the integration approximation leads to the collocation method of RBF for Poisson's equation. Next, the combinations of RBF with finite-element method (FEM), finite-difference method (FDM), etc., can be easily formulated by following Li [1] and Hu and Li [2,3], but more analysis of inverse estimates is explored in this paper. Since the RBFs have the exponential convergence rates, and since the collocation nodes may be scattered in rather arbitrary fashions in various applications, the RBF may be competitive to orthogonal polynomials for smooth solutions. Moreover, for singular solutions, we may use some singular functions and RBFs together. Numerical examples for smooth and singularity problems are provided to display effectiveness of the methods proposed and to support the analysis made.