Partial Differential Equations Grid Domain Decomposition Method

In this thesis, we will combine the meshless methods using radial basis functions(RBF) with non-overlapping domain decomposition methods(DDM) to solve elliptic partial differential equations of second order. The traditional non-overlapping DDM include Dirichlet-Neumann iteration, Neumann-Neumann iteration, Robin iteration et.al. The meshless method using RBF insist of collocation method and Galerkin methods. Compared with Galerkin method, collocation method is higher performance in numerical computation. But the theoretical analysis of collocation method is quite absence. For example, we can not to be sure that the collocated matrix is invertiable. On the other hand, RBF Galerkin method can not carry out for Dirichlet boundary value problems(BVP). These problems will arise from either Dirichlet-Neumann iteration or Neumann-Neumann iteration. We will use the method provided by Cai to deal with it. Also, we will discuss the Lagrange multipliers method to conquer it. Based on the analysis for the combination of non-overlapping DDM and RBF meshless method, we find that the convergent rate of meshless Dirichlet-Neumann iteration dramatically depend on the density h, while the Robin iteration is lack of information about the convergent rate. So, we hope that there is some DDM, which need not the iteration and Dirichlet BVP, if we consider the Helmholtz equation with Neumann boundary conditions. Moreover, the method should include the information about convergent rate. For these requests, we will consider the projection DDM, which will be an alternative perspect opposed to the projection DDM based on spectral collocation methods or finite element methods. The main contents of this thesis include:1. We will deal with the essential BVP by Lagarange multipliers. Error estimate and numerical example will be given.2. we will give the estimate of condition number of the collocation methods using RBF.3. we will combine the non-overlapping DDM with RBF meshless method, including collocation method and Galerkin method. The convergence analysis and numerical examples will be given. We just consider Dirichlet-Neumann iteration and Robin iteration.4. Based on the analysis above, we will combine projection domain decomposition method...