Local And Parallel Finite Element Method For Laplace Eigenvalue Problem

The local and parallel finite element discretization method was first established by Xu and Zhou. In recent years, many researchers have been developing the local and parallel finite element method, which has become an important and efficient numerical method for solving partial differential equations. Based on their work[6], this paper establishes the local and parallel finite element method for solving the Laplace eigenvalue problem with Dirichlet boundary conditions. Theoretical analysis and numerical experiment are given in this paper to verify the efficiency of the scheme.The technique used in this paper are parallel algorithms, which is based on the twogrid discretization and defect-correction on local meshes, is very important methods to reduce the computational costs and improve the accuracy of finite element solutions. With this technique, this method solve eigenvalue problem on a fine grid, reduced to solving a boundary value problem on coarse grid, a boundary value problem on mesoscopic grid and solve a few boundary value problems on local areas in parallel. On the other hand,it was further investigated and successfully used in other problems, such as Stokes equations. Absolutely, this technique also can solve self-adjoint elliptic differential operation eigenvalue problems.In this paper, we combine the finite element method, the local and parallel algorithms and Rayleigh quotient iteration to solve the Laplace problem. Finally, we perform experiments by Matlab software and compare our results with two algorithms respectively, as results show, our method is satisfactory.