The Partition Of Unity Method On The Basis Of Local Polynomial Approximation Spaces And Its Error Analysis

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. These methods are designed to handle more effectively problems and other difficult problems. Partition of unity method is one 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 method can therefore be more efficient than the usual finite element methods.This paper consists of three chapters. In the first part, we present mathematical foundations of the partition of unity method. Local approximation spaces are choosen according to the problems. Here we analyse in more detail the partition of unity method based on local polynomial approximation spaces, and the local basis functions (multiplied by the appropriate partition of unity functions) may be linearly dependent and thus do not form a basis of the partition of unity space. Subsequently we present local approximation spaces of some interpolation polynomial.In the second section, we begin with the error of the finite element method, then analyse a special partition of unity method (the basis functions of finite element method form a partition of unity). We have proved the order of it's error is higher O(h¹) than that of local space in partition of unity method. The results we obtained are accurate to local space of any degree. And the same time, they are accurate for L₂-norm and L∞-norm.In the third section, combining these results, we obtain general conclusions for the model problems and also make some comments on the prospect of the error analysis.