Abstract: 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.

Key words: finite point method, directional differential, directional difference