The Coupling Method And Domain Decomposition Method For Some Nonlinear Problems

In this thesis, by the theory of the natural boundary reduction and the key idea of domain decomposition method for the exterior problems, we study some numerical techniques for several kinds of nonlinear problems.In Chapter1, based on the Kirchhoff transformation and the natural boundary element method, we investigate a coupling of the finite element method and the nat-ural boundary element method for quasilinear problems in a bounded or unbounded domain with a concave angle. By the principle of the natural boundary reduction, we obtain the natural integral equation on circular arc artificial boundaries, and get the coupled variational problem and its numerical method. Moreover, the convergence of approximate solutions and error estimates are obtained. Finally, some numerical examples are presented to show the feasibility of our method.In Chapter2, based on the Kirchhoff transformation, the coupling of finite ele-ment method and natural boundary element method is discussed for solving quasilin-ear exterior anisotropic problems with elliptic artificial boundary. By the principle of the natural boundary reduction, we obtain natural integral equation on elliptic artificial boundaries, the coupled variational problem and its numerical method. Moreover, the convergence and error estimates of the approximate solutions are obtained. Finally, some numerical examples are presented to illustrate the feasibility of the method.In Chapter3, a Dirichlet-Neumann (D-N) alternating algorithm which is a non-overlapping domain decomposition method, based on natural boundary reduction principle, is presented for solving quasilinear exterior anisotropic problems with ellip-tic or circular artificial boundary. By the principle of the natural boundary reduction, we obtain the natural integral equation for the anisotropic quasilinear problem on el-liptic and circular artificial boundaries and construct the algorithm and analyze its convergence. Moreover, we give the existence and uniqueness result for the original problem. Finally, some numerical examples are presented to illustrate the feasibility of the method.In Chapter4, the artificial boundary condition, derived in terms of infinite Fourier series, is applied to solve a class of quasi-Newtonian Stokes flows. Based on the natural boundary reduction, we obtain the artificial boundary condition on the artificial boundary, the coupled variational problem and its numerical method. We also provide the unique solvability of the continuous cases and discrete formulations and study a priori error estimate error analysis of the problem. At last, we provide a posteriori error estimate for the corresponding problem.In Chapter5, we discuss a coupling of natural boundary element and finite el-ement method for a quasilinear incompressible elasticity problem in an unbounded domain. Based on the Kirchhoff transformation and the principle of the natural bound-ary reduction, we obtain the natural integral equation for the quasilinear problem on circular artificial boundaries and the coupled variational problem. What’s more, we shall discuss the unique solvability of the problem, derive a Cea-type estimate for the associate error and show that the error depends on the approximate artificial boundary condition and the location of the artificial boundary.