Some Domain Decomposition Methods For Exterior Three-Dimensional Problems With Ellipsoidal Artificial Boundary

The theory of the natural boundary reduction (NBR), suggested by Prof. K. Feng, is a distinctive one among various boundary reductions. It is the third major academic contribution of Prof. Feng besides the finite element method and the sym-plectic algorithm. Later Prof. D. Yu has done a lot of important work in this field. Except that the direct natural boundary element method can be used to deal with some boundary value problems over some special domains, while the coupling method and domain decomposition method (DDM) based on the NBR become two of the efficient methods in solving problems over unbounded, concave or cracked domains. Up to now, a circle (2-D) or a sphere (3-D) was usually chosen as an inner boundary of an exterior unbounded domain, and the research results are much rich. However, using an ellipsoid as an artificial boundary for some special inner domains, for example using prolate ellipsoid for cigar-shaped domains, can lead to a less computational domain, and as a result reduce computational cost and memory capacity.In this thesis, the DDMs based on the NBR for some exterior3-D Poisson prob-lems are investigated. In chapter1, we introduce two kinds of ellipsoid coordinates, some special functions and some Sobolev spaces, which are important theoretical ba-sics of the thesis. In chapter2and chapter3, we consider the3-D Poisson equation over unbounded domains. Two kinds of ellipsoids (prolate ellipsoid and general el-lipsoid) are chosen as an artificial boundary, D-N alternating algorithms based on explicit two kinds of ellipsoids and exact artificial boundary conditions are presented. The convergence of the discrete D-N alternating algorithm is analyzed, the error es-timates of the numerical solutions are given, and some numerical examples are pre-sented to illustrate the feasibility of the method. In chapter4, a Schwarz alternating algorithm based on natural boundary reduction for the exterior3-D harmonic problem is discussed, the convergence and the rate of convergence of the algorithm is analyzed, the error estimates of the numerical solutions are given, and some numerical examples are presented to illustrate the feasibility of the method.