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.

研究重点是具有高阶逼近方案的无网格有限差分法的发展及其在椭圆问题中的应用。与有限元方法和其他无网格方法相反，可以在不引入任何新节点或自由度的情况下提高近似阶数。主要概念基于对差分算子的附加校正项的考虑，通过移动加权最小二乘法技术生成。包含在这些术语中的高阶导数使用基本离散化和近似模型进行评估，即通过适当的公式组合和主要数值解。修正项仅修改迭代求解的代数方程的右侧。以这样的方式，避免了病态和奇异有限差分方案的问题。这种技术可以应用于在局部（强）和全局（弱变分）公式中提出的椭圆问题。

该论文用选定的一维和二维基准椭圆问题的结果进行说明，具有各种几何形状以及几个工程应用。注意力集中在解决方案及其导数的准确性上，以及在规则网格集和节点的不规则云集上估计的收敛率。此外，还考虑了基于高阶解的后验误差估计。