Anisotropy Outside The Problem Domain Decomposition Algorithm

In this paper, through the idea of domain decomposition method and the theory of natural boundary reduction, domain decomposition methods based on the natural boundary reduction on an elliptic boundary, which includes overlapping and non-overlapping domain decomposition methods, are investigated for a kind of anistropic elliptic problem with constant coefficients in an exterior domain. For non-overlapping domain decomposition method, we prove the convergence of the algorithm, and equivalence to the Richardson iterative algorithm. Discrete D-N algorithm is given. The choice of the relaxation factor is analyzed in detail. Some numerical experiments are presented. For overlapping domain decomposition method, we prove the geometric convergence for continuous algorithm in the sense of energy norm by means of projection theory, analyze the convergence rate of the algorithm in detail. And we also discuss the discretization of the algorithm and its finite element implementation. Some numerical experiments are also given. Theoretical analysis as well as numerical results show that our methods are high performance for solving anisotropic exterior elliptic problems.