Domain Decomposition Methods For Exterior Helmholtz Problems

The natural boundary reduction(NBR), suggested by Professor Feng Kang, is a particular one among various boundary reductions. It is the third major academic contribution of Professor Feng Kong beside the finite element method (FEM) and the syntectic algorithm. Later Professor Yu Dehao has done many important work in this field. At present, the direct natural boundary element method(NBEM) can be used to deal with some boundary value problems over several special domains, while the domain decomposition method(DDM), which is based on the NBR, become one of the efficient methods in solving problems over unbounded, concave or cracked domains. Up to now, the above work has been done for some 2-D and 3-D problems, and the research results are much rich. A circle or spherical surface is usually chosen as an artificial boundary. However, using elliptic or ellipsoid as an artificial boundary for some special shaped inner domains, such as elliptic or ellipsoid domains, can lead to a less computational domain and as a result reduce computational number and memory capacity. This thesis is mainly aimed to study the DDM (based on the NBR) for solving 3-D exterior Helmholtz problems, and a Dirichlet-Neumann (D-N) alternating algorithm for the time-dependent anisotropic problem in an exterior elliptic domain.In chapter 1, some ellipsoid coordinates、special functions、Sobolev spaces and some important theorems are introduced. These contents are important theoretical bases of this thesis and will be frequently referred in the following chapters.In chapter 2, we study a D-N alternating algorithm based on the NBR for solv-ing 3-D exterior Helmholtz problems. Firstly, the D-N alternating algorithm and the Richardson iterative algorithm which is equivalent to the original method are given. Secondly, the convergence of algorithm is analyzed, and the variational form and dis-cretization are obtained, and the convergence of the discrete variational form is also analyzed. Finally, some numerical examples are presented to illustrate the feasibility and efficiency of this method.In chapter 3, we study the Schwarz alternating algorithm based on the NBR for solving 3-D exterior Helmholtz problems. Firstly, the variational form which is e-quivalent to the original problem is given. Secondly, the Schwarz alternating algo- rithm based on the NBR is suggested, the convergence and the convergence rate of algorithm are analyzed, and find out the relationship between contraction factor a and overlapping degree. Besides, some error estimates of the algorithm are obtained. Fi-nally, some numerical examples are presented to illustrate the feasibility and efficiency of this method.In chapter 4, we study a D-N alternating algorithm based on the NBR for solv-ing 3-D exterior Helmholtz problems with prolate spheroid boundary. Firstly, a D-N alternating algorithm based on NBR is suggested. Then the convergence of the algo-rithm is analyzed.In chapter 5, we study a D-N alternating algorithm for the time-dependent anisotropic problem in an exterior elliptic domain. Using the transformation of vari-ables, the original problem is changed into a Helmholtz problem. By means of the theory of the NBR, the Poisson integral formula and the natural integral equation are obtained. Secondly, a D-N alternating algorithm based on the NBR is suggested, and the convergence of algorithm is analyzed. Finally, some numerical examples are presented to illustrate the efficiency of our method.