Domain Decomposition Method For The Stokes Equation On Unbounded Domains

The Stokes problem is a classical problem in computational fluid dynamics. It is the boundary value problem of a type of partial differential equations derived from fluid dynamics, which is widely used in many fields, such as chemical engineering, environmental engineering, mineral processing, grouting and so on. Through studying Stokes problem, we can solve stationary problem of viscous incompressible homogeneous fluid flow, also we can lay a foundation for dealing with the more complicated Navier-Stokes problems. So, it is great significant to do research on the Stokes problem for the development of fluid dynamics. For the complexity of Stokes problem, and the problems of complex boundary conditions, it is much difficult to obtain the analytical solution of Stokes problem. Consequently, computer is used to study its numerical solution.Risen in the 1980's, domain decomposition method is a efficient numerical method for solving partial differential equations. The calculation region is divided into several subregions, the global approximation solution is obtained through the transformation from original problem to simple problems in subregions and iteration method. Although domain decomposition method is highly effective for the Stokes problem on bounded domains, when come to unbounded domains, this algorithm meet essential difficulties. Natural boundary reduction is a very powerful tool to solve differential equations in unbounded domains. So, in this paper, the the Stokes problem in unbounded domains is solved by domain decomposition method based on natural boundary reduction, this method has the advantages of natural boundary reduction and the flexibility of domain decomposition method. The saddle point problem is derived from the discrete form of Stokes problem. Uzawa type algorithm is an efficient iteration algorithm to solve saddle point problem, which is special suitable for the computation of large sparse matrix.For domain decomposition method, in this paper, a non-overlapping domain decomposition method is investigated for solving the exterior boundary value problems of the Stokes equation. The convergence analysis for the D-N alternating algorithm (including continuous and discrete form) is presented, also the necessary and sufficient condition for convergence of this method is gained, then the optimal relaxation factor is obtained, finally some numerical results are illustrated.For Uzawa type algorithms, in this paper, the necessary and sufficient condition for convergence of preconditioned Uzawa algorithm is gained, also the sufficient condition is presented and spectral radius of error transfer matrix is gained. The necessary and sufficient condition for convergence of inexact Uzawa algorithm is wholly proved. Based on a new norm, a weaker sufficient condition for convergence of nonlinear inexact Uzawa algorithm is given, and also the estimation of convergence rate is obtained. Finally the discrete form of the the Stokes problem is solved by Uzawa type algorithms and some numerical results are illustrated to verify the theory.