On The Domain Decomposition Method For Some Partial Differential Equations Based On The Natural Boundary Reduction

Both the boundary element method (BEM) and the finite element method (FEM) are recognized now as general numerical methods which are applicable to a wide variety of engineering problems. In general, the BEM is more suitable for problems over unbounded domains, while it is usually confined to the region with homogeneous material, where the governing equations are linear. On the other hand, the FEM is restricted to a bounded region, whereas it is applicable to problems where the material properties are not necessarily homogeneous and nonlinearities may occur. The natural boundary element method (NBEM) not only has those advantages that other boundary element methods possess, but also has some distinctive advantages.It is always paid close attention to solving the boundary problems of partial differential equation (PDE) over unbounded domains. Lots of numerical methods were tried to deal with difficulty arised from the infinity of the domains. On the other hand, domain decomposition method (DDM) has been as important focus in the field computational mathematics. We study a kind of numerical method to solve the elliptic problems over unbounded domains by means of overlapping and non-overlapping DDM.First we investigate an overlapping domain decomposition method based on the natural boundary reduction on elliptic boundary for the anisotropic elliptic PDE with constant coefficients in an unbounded domain.Second we combine the generalized difference methods on CC dual subdivision with the natural boundary element method to solve a kind of anisotropic and semilinear elliptic PDE in an unbounded domain.In chapter 1, we mainly introduce the contents of research, the meaning of research,the status of recent researches , the tendency of development and some basic knowledge of finite element, boundary element and the generalized difference methods. The contents are important theoretical bases of this paper and will be reffered in the following chapters. In chapter 2, we investigate an overlapping domain decomposition method based on the natural boundary reduction on elliptic boundary for the anisotropic elliptic PDE with constant coefficients in an unbounded domain. We prove its geometric iterative convergence with maximum norm in the continuous case and obtain an optimal iteration convergence factor, which is independent of the anisotropic degree, by using Fourier analysis with confocal elliptic boundaries. We also prove its geometric convergence in the discrete case and obtain the error estimate of the iterative convergent solution by using the maximum principle. Finally, our numerical results confirm the theoretical convergence analysis and show the advantage for solving the anisotropic elliptic PDE in unbounded domains. In chapter 3, we combine the generalized difference Methods with the natural boundary element method to solve a kind of anisotropic and semilinear elliptic PDE in an unbounded domain. We carry out discretization to obtain the difference scheme and the nonlinear equations by the generalized difference method. By the error estimate theory of the finite element and the natural boundary element, we obtained the error estimate and the asymptotic rate of on convergence O(h) using the interpolation theory of the generalized difference method.