Application Of RBF Collocation Method And Its Error Estimate

Meshless method is a new numerical method on problem for determining solution of partial differential equation. It is an approximation method based on nodes, and doesn't need a mesh entirely or partly. Then the discontinuous problems, the extremely large deformations problems, and movable boundary problems, etc, can be solved by use to meshless without the re-meshing techniques. And not only the calculated precision can be guaranteed, but also the calculated difficulty can be decreased. In this paper, after introducing the meshless method simply, the interpolation process of the compactly supported radial basis functions is droved. But the compactly supported radial basis functions can not reconstruct constant function and k-order polynomial, which can not embodies the compact predominance. For improving the precision, we drive the consistent compactly supported radial basis functions are droved farther, which satisfy the consistency conditions. Then, the nonlinear Poisson equation and the plane piezoelectric structures using collocation approach method with radial basis functions (RBF) are analyzed. The validity and accuracy of RBF collocation method are verified through numerical examples. Numerical results of the different radial basis functions are compared. The factors of calculating accuracy are analyzed. Finally, the common process and conclusions of exponential type error estimate on RBF collocation method are given. The error estimates of nonlinear Poisson equation and plane piezoelectric structure are obtained using RMQ (Reciprocal multiquadrics) basis function. It is respectively O (λc /h) and O (λc /h).