Weighted radial basis collocation method for boundary value problems

Abstract

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.

Convergence Studies for an Adaptive Meshless Least-Squares Collocation Method

“…Using reproducing kernels of the Sobolev space H m (Ω), we proved that the weighted least square solution converges to the analytical solution in an optimal rate h m-2 in H 2 -norm. Numerical evidence in two dimensions also showed that the weighted least square problem gives more accurate results than the unweighted one [12].…”

“…Using reproducing kernels of the Sobolev space H m (Ω), we proved that the weighted least square solution converges to the analytical solution in an optimal rate h m-2 in H 2 -norm. Numerical evidence in two dimensions also showed that the weighted least square problem gives more accurate results than the unweighted one [12].So far, we did not address the problem of ill-conditioning, which depends on the choice of the kernel, its shape parameter, data points distribution, and literally everything in the partial differential equation. One solution is to look for a well-behaved subspace of the trial space so that the condition of the reduced linear system can be controlled.…”

A collocated C0 finite element method: Reduced quadrature perspective, cost comparison with standard finite elements, and explicit structural dynamics

“…The main advantage of collocation as compared to Galerkin methods is a significant reduction of the computational cost for the formation and assembly 1 of stiffness matrices and residual vectors [13][14][15][16]. Collocation methods attracted wide attention during the 1970's and 80's (see for example [9,10,[13][14][15][17][18][19]), and have been widely applied in spectral element methods [1,2,11,12,20], in meshfree methods [21][22][23][24][25][26], and most recently in isogeometric analysis [16,[27][28][29][30].…”

A Cartesian‐grid collocation technique with integrated radial basis functions for mixed boundary value problems

“…Pan et al [17] presented meshless Galerkin least-squares method by making use of Galerkin method in the boundary domain and least-squares method in the interior domain. Hu et al [18,19] introduced the weighted radial basis collocation method in which the residual error on the Neumann boundary is treated by a proper scaling weight. Atluri et al [20,21] proposed a "mixed" collocation technique, however, stable solutions are obtained with resort to the local weak form at nodal points for stress and the penalty method for Neumann boundary conditions.…”