Decomposition Algorithm, Based On The Natural Boundary Reduction Of The Semi-unbounded Region Region

In this paper, based on the theory of natural boundary reduction, the numerical resolution methods for elliptic boundary value problems on a kind of unbounded half plane with a cave are discussed. In the frame of the domain decomposition method, the domain is decomposed into a bounded sub-domain to which the finite element method is applied and a regular unbounded sub-domain to which the natural boundary reduction is applied. So iterative algorithms are constructed on both of the sub-domains. First overlapping domain decomposition is considered, to which the Schwarz alternative method is applied. By P. L. Lions' projection explanation of the Schwarz alternative method it is proved that the convergence rate of the algorithm is geometrical. Then non-overlapping domain decomposition is considered, to which the Dirichlet-Neumann method is applied. Both the equivalence to the Richardson iterative method and the convergence rate independent of the finite element mesh size of the algorithm are proved. With proper selection of the relaxation factor it is proved that the convergence of the D-N alternative method is geometrical and the general values of the relaxation factor are given.