Weighted radial basis collocation method for boundary value problems

This work introduces the weighted radial basis collocation method for boundary value problems. We first show that the employment of least-squares functional with quadrature rules constitutes an approximation of the direct collocation method. Standard radial basis collocation method, however, yields a larger solution error near boundaries. The residuals in the least-squares functional associated with domain and boundary can be better balanced if the boundary collocation equations are properly weighted. The error analysis shows unbalanced errors between domain, Neumann boundary, and Dirichlet boundary least-squares terms. A weighted least-squares functional and the corresponding weighted radial basis collocation method are then proposed for correction of unbalanced effors. It is shown that the proposed method with properly selected weights significantly enhances the numerical solution accuracy and convergence rates.