有限点方法研究

在二维散乱离散点集上研究一类无网格方法——有限点方法(Finite Point Method,简称FPM),建立方法的基础.采用方向微商和方向差商讨论有限点方法,建立各阶各方向微商间的关系式.利用这些关系式,根据被逼近点的邻点数目差异,分别建立数值方向微商的五点公式及少点(两点、三点、四点)公式;研究五点公式的可解性条件与可允许邻点集;获得典型微分算子的数值方向微商公式等.理论分析和数值试验表明,随着邻点数目的增加,相应数值公式的逼近精度随之提高.这类近似公式不仅为在散乱离散点集上构造各类偏微分方程的格式奠定了基础,同时,也可应用于偏微分方程非结构网格计算方法,提高方法的精度.

Study on Finite Point Method

A class of meshfree methods——finite point method on a set of two-dimensional disordered points is studied. Fundamentals of the method are established by means of directional differentials and directional differences. Formulae relating to multi-directional differentials of each order are given. Based on these formulae and with different numbers of neighboring points, five-peint formulae and less-point (two-point, three-point and four-point) formulae are derived, respectively. Solvability conditions of the five-point formulae and permissible set of neighboring points are discussed. Approximate expressions for classical differential operators on a set of disordered points are derived. It is demonstrated with theoretical analysis and numerical experiments that the accuracy of these formulae is improved as the number of neighboring points increases. These approximate formulae lay foundation for constructing computational schemes of partial differential equations on a set of disordered points. They can be applied to computational methods on unstructured meshes to increase accuracy as well.