Error Analysis Of The Partition Of Unity Method Based On Local Polynomial Approximation Space

Recently, Meshfree methods have attracted much interest in the scientific computation. This new family of numerical methods shares a common feature that no mesh is needed. Those methods are designed to handle more effectively problems and other difficult problems. Partition of unity methods is a very important kind of the meshfree methods.The obvious feature of it is having the ability to include in the finite element space knowledge about the partial differential equation being solved.This methods can therefore be more efficient than the usual finite element methods. I. Babuska, J. M. Melenk and many other scholars have made great efforts and PUFEM made a great progress. But until now I. Babuska can not get optimal order error estimates for PUFEM interpolants. The goal of this paper is to get optimal order error estimates for PUFEM interpolants by choosen of a kind of special polynomial local approximation space.In this paper,Firstly we present mathematical foundation of the partition of unity methods.Local approximation spaces are choosen according to the prob-lems.Subsequently we present local approximation spaces of some interpolation polynomial.Professor. YunQingHuang ,WeiLi,FangSu had got optimal order error estimates for PUFEM interpolants on the basis of a kind of special polynomial local approximation space in one dimensional case. In the second section, We will list relevant results.In the last section We construct a kind of special polynomial local approximation space in quadrilateral element of two dimensionals. and derive optimal order error estimates for PUFEM interpolants .Finally combining these results, we obtain general conclusions for the model problems and also make some comments on the prospect of the error analysis.