Higher order schemes introduced to the meshless FDM in elliptic problems

Abstract

The research is focused on the development of the Meshless Finite Difference Method with higher order approximation schemes and its application in elliptic problems. On the contrary to the Finite Element Method and other meshless methods, the approximation order may be raised without introducing any new nodes or degrees of freedom. The main concept is based upon the consideration of additional correction terms of difference operators, generated by means of the Moving Weighted Least Squares technique. Higher order derivatives, included in those terms, are evaluated using basic discretization and approximation models, namely by the appropriate formulas composition and the primary numerical solution. Correction terms modify only right–hand sides of algebraic equations, which are solved iteratively. In such a manner, problems with ill–conditioned and singular finite difference schemes are avoided. This technique may be applied to elliptic problems posed in both local (strong) and global (weak variational) formulations.The paper is illustrated with results of selected one and two dimensional benchmark elliptic problems, with various geometrical shapes as well as several engineering applications. The attention is laid upon the accuracy of solution and its derivatives as well as convergence rates estimated on the set of regular meshes and irregular clouds of nodes. Moreover, a posteriori error estimates, based upon higher order solution, are taken into account.