SUMMARYThis 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 errors. It is shown that the proposed method with properly selected weights significantly enhances the numerical solution accuracy and convergence rates.

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The earlier work in the development of direct strong form collocation methods, such as the reproducing kernel collocation method (RKCM), addressed the domain integration issue in the Galerkin type meshfree method, such as the reproducing kernel particle method, but with increased computational complexity because of taking higher order derivatives of the approximation functions and the need for using a large number of collocation points for optimal convergence. In this work, we intend to address the computational complexity in RKCM while achieving optimal convergence by introducing a gradient reproduction kernel approximation. The proposed gradient RKCM reduces the order of differentiation to the first order for solving second-order PDEs with strong form collocation. We also show that, different from the typical strong form collocation method where a significantly large number of collocation points than the number of source points is needed for optimal convergence, the same number of collocation points and source points can be used in gradient RKCM. We also show that the same order of convergence rates in the primary unknown and its first-order derivative is achieved, owing to the imposition of gradient reproducing conditions. The numerical examples are given to verify the analytical prediction.convergence in RBCM [25-27], the linear system of RBCM is typically ill-conditioned [28,29]. An alternative approach is the employment of smooth approximation with compact support such as the MLS or RK approximation in the strong form collocation method [14,18,19,30,31]. The reproducing kernel collocation method (RKCM) offers a much better conditioned discrete system than that of RBCM; nevertheless, it converges algebraically [30,31]. The work in [32] shows that one can construct a localized RBF using a partition of unity function, such as the reproducing kernel enhanced radial basis function, to yield a local approximation while maintaining the exponential convergence in RBCM. This localized RBF, combined with the subdomain collocation method, has been applied to problems with local features, such as problems with heterogeneity [33] or cracks [34] that are difficult to be solved by RBCM.It is noteworthy that higher order derivatives of the approximation functions are needed in the strong form collocation method compared with the Galerkin method. While approximation functions such as RK and MLS can be arbitrarily smooth, taking derivatives of these functions is computationally costly, making RKCM less efficient. In particular, the high complexity in RKCM is caused by taking derivatives of the moment matrix inversion in the multidimensional RK shape functions (see the detailed complexity and error analysis of RKCM in [31] and [30], respectively). Furthermore, for optimal convergence in RBCM and RKCM, using the number of collocation points much larger than the number of source points is needed, and this adds additional computational effort [15,30]. Motivated by the above mentioned disadvantages in RKCM, a gradient RK approxim...

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