On the Convergence of Iterative Methods and Pseudoinverse Approaches in Global Meshless Collocation

Abstract

In the context of numerical approximation of partial differential equations, meshless methods are recent developments, which have been reported to be relatively straightforward, yet provide better convergence and accuracy as compared to the conventional mesh-based approaches, for some specific problems like stress–strain analysis and modeling in a deforming media. Among several proposed schemes, strong-form collocation schemes using radial basis functions are easy-to-program, require no mesh in the domain or at the boundary, avoid numerical integration, and have similar formulations for all dimensions. Some of the radial basiss functions like the Gaussian, multiquadric and inverse-multiquadric are infinitely smooth. The standard global meshless formulations based on such radial basis functions are often found to lead towards ill-conditioned systems of linear equations. For such formulations, we investigate the degree of ill-conditioning against degrees of freedom for different radial basis functions. Also, the convergence of four iterative methods and pseudoinverse approaches has been tested for the solution of such ill-conditioned systems. In order to compute the pseudoinverse, two different approaches, i.e., singular value decomposition and full rank Cholesky factorization have been used. A set of numerical experiments, performed here, demonstrate that pseudoinverse approach based on singular value decomposition performs better than the iterative methods. Although, pseudoinverse computation via full rank Cholesky factorization is faster, it is not recommended in global meshless collocation schemes due to poor convergence at relatively large degrees of freedom.