A Dirichlet-Neumann Alternating Algorithm For Two-dimensional Kardar-Parisi-Zhang Equation In An Unbounded Domain

In this thesis, by the theory of the natural boundary reduction and the key idea of domain decomposition method (DDM), a Dirichlet-Neumann (D-N) alternating algorithm based on the natural boundary reduction is devised for the solution of the Kardar-Parisi-Zhang (K-P-Z) equation in an exterior two-dimensional domains. The main work of this thesis is as follows.Firstly, based on the Cole-Hopf transformation, the K-P-Z equation is changed into a heat conduction equation, and the governing equation is dis-credized in time by the Newmark method, leading to a time-stepping scheme, where a exterior elliptic problem has to be solved at each time step. Secondly, a circular artificial boundary is introduced, the Poisson integral formula and the natural integral equation of the problem in exterior circular domain by the prin-ciple of the natural boundary reduction (NBR). Thirdly, by means of the results of NBR, the Dirichlet-Neumann alternating algorithm for solving the discrete problem is proposed. the convergence of the algorithm is analyzed, and the con-vergence rate is independent of the size of finite element mesh. And it is proved that the D-N alternating algorithm is equivalent to preconditioned Richardson iteration method. Finally, some numerical experiments are presented to illus-trate the feasibility and effectiveness of the method.