Optimal Error Analysis Of Partition Of Unity Method And Application Of Algebraic Multigrid Algorithm

This paper consists of two parts.In the first part,we provide optimal error estimates of a class of partition of unity method(PUM) with local polynomial approximation spaces.Recently,meshfree methods have attracted much interest in the scientific computation.In contrast to classical FEM,this new family of numerical methods shares a common feature that no mesh is needed.Those methods are designed to handle more effectively problems in complex domains,or in domains evolving with the problem solution.The partition of unity method is one of important meshfree methods.One of the important aspects of PUM is that it permits the use of partition of unity functions,whose supports may not depend on any mesh(e.g.Shepard functions,see[4]),or may depend on a simple mesh that does not conform to the geometry of the domain.In this sense,the PUM is also a meshless method and this feature allows us to avoid the use of a sophisticated mesh generator.Another important aspect of PUM is that local approximation spaces can have functions other than polynomials,which locally approximate the unknown solution well.Hence PUM made a great progress.There are many recent papers on PUM,but most of them are of engineering character,without any mathematical analysis.I.Babu(?)ka and his co-workers did much fundation work.But until now I.Babu(?)ka can not get optimal order error estimates for PUM interpolants in his papers.The goal of this part is to get optimal order error estimates for PUM interpolants by chosen of a kind of special polynomial local approximation space.For this purpose,we construct a special polynomial local approximation space according to the consistence and local approximation properties of PUM at first.Then the PUM interpolation scheme of higher degree in 1D and the PUM interpolation scheme of lower degree in 2D are given,and optimal error estimates for PUM interpolants are derived.The interpolation error estimates are used to obtain optimal order error estimates for Galerkin solution in 1D.In the second part,a modified adaptive algebraic multigrid algorithm for a class of elliptic variational inequalities and parallization are investigated.According to the linear complementarity of discrete elliptic variational inequalities,a modified algebraic multigrid(AMG) algorithm based on an active-set strategy is presented to solve the discrete problems of variational inequalities with symmetric second-order elliptic operators.The numerical experiments show the efficiency and robustness of the proposed algorithm both on the uniform mesh and on h-adaptive mesh.To shorten computation time,this part presents a parallel scheme for the modified adaptive AMG.Numerical experiments illustrate the speedup and efficiency of the parallel scheme.