The Applications Of The Natural Boundary Element Method And Domain Decomposition Method In Some Nonlinear Problems

Finite element method (FEM) and boundary element method (BEM) are commonly used numerical method for solving many engineering problems. BEM is applicable to lin-ear, homogeneous and unbounded region, but restricted by the complexity of the problem and the region. BEM is suitable for solving nonlinear and inhomogeneous problems in bounded areas. Natural boundary element method (NBEM) is a kind of BEM which is put forward by the Chinese scholars for the first time, this method not only has the common characteristics of the BEM, but also has many unique advantages. The FEM and NBEM can be coupled naturally because they base on the same variational principle.Domain decomposition method (DDM) is a new technology for numerical solving par-tial differential equations. It can decompose a big problem into smalls, a complex boundary value problem into simples, and a serial problem into parallels. With the rapid development of parallel computers, it has become a hot area of computational mathematics. It allows to set up different mathematical models, to choose different calculation methods and different mesh subdivisions in different sub areas, so compared with other methods, it is more flexi-ble. When the problem in unbounded region is divided into several sub areas, at least one is in unbounded region, at this point, it is a good choice to deal with the subproblem by BEM (or coupling method with FEM).The main work of this paper is using NBEM and DDM to solve nonlinear problems. In the first part, we mainly consider a class of nonlinear transmission problems with Coulomb friction. First of all, we present a new coupling framework of finite element and natural boundary element (FEM-BEM) by introducing a circular artificial boundary, and prove the existence and uniqueness of the solution of the discrete coupled problem. New coupling framework can avoid solving boundary integral equations and just calculate some singular integrals in the process of discretization. Second, we give the error estimate of solution of discrete problem. Finally, we present a new iterative algorithm for the coupling framework. The algorithm makes the finite element and boundary element part (instead of a small finite element problem) independently solved. Moreover, we strictly prove that the convergence rate of the iterative algorithm is independent with the mesh size, and give the optimal param-eter of iterative algorithm. A numerical example validates the effectiveness of the iterative algorithm, in other words, our algorithm is effective.In the second part, we mainly study a class of nonlinear problems. Some simple models of the problem have been studied in [1-5]. This paper generalizes the results of [6], and explores a new algorithm. We present the error estimate of the solution of discrete problem, design a numerical example on general area, to verify the validity of the theoretical analysis. We find that the discrete variational problem is difficult to solve as a whole. The stiffness matrix of the natural boundary integral operator is dense, thus caused the stiffness matrix of the whole problem is dense, so it brings the expensive computing. To solve the problem, we propose a domain decomposition algorithm which is based on natural boundary reduction.