The Study Of Algorithms For Several Problems Based On Natural Boundary Reduction

The natural boundary reduction (NBR), suggested firstly by Prof. Feng Kang, is a distinctive one among various boundary reductions. It is one of the three major aca-demic contribution of Prof. Feng besides the finite element method and the symplectic algorithm. Later Prof. Yu Dehao has done varieties of important work in this field.The natural boundary element method (NBEM) can be used to deal with some bound-ary value problems over some special domains directly, also the coupling method,domain decomposition method (DDM) and multigrid algorithm (MGA) based on the NBR become the efficient methods in solving problems over unbounded, concave or cracked domains. Up to now, the circle (2-D) or sphere (3-D) are usually chosen as ar-tificial boundaries of an exterior unbounded domain, and the research results are much rich. However, using elliptic or ellipsoid as artificial boundaries for some special inner domains, as strip or cuboid domains, can lead to a less computational domain, and as a result of reducing computation cost and memory capacity. This thesis mainly focuses on study the DDM (Schwarz and D-N alternating method) based on NBR for solv-ing 3-D exterior anisotropic problems, and MGA based on NBR for 2-D Helmholtz problems in an infinite domain.In chapter 1, we introduce two kinds of orthogonal coordinate systems, some special functions, some Sobolev spaces and some important theorems therein, which are necessary theoretical basis of the thesis.In chapter 2, we consider the Schwarz alternating algorithms based on the NBR for solving 3-D anisotropic problems over unbounded domains. Using transformation of variables, the original problem is changed into a Laplace problem. Based on the NBR, we obtain the natural integral equation and Poisson integral formula. Then, a Schwarz alternating algorithm on the NBR is proposed,and its convergence is ana-lyzed. Finally, some numerical examples are computed to illustrate the feasibility and efficiency of our method.In chapter 3, we study the D-N alternating algorithms based on the NBR for solving 3-D exterior anisotropic problems. Based on chapter 2, D-N alternating algo-rithm based on the NBR and the Richardson iterative algorithm which is equivalent to the original method is suggested. Secondly, the convergence of the algorithm is analyzed, and the equivalent variational form and discretization are obtained. Finally,some numerical examples are presented to illustrate the feasibility and efficiency of the method.In chapter 4, a multigrid algorithm based on the NBR for the exterior 2-D Helmholtz problems is discussed. Firstly, the equivalent variational form of the orig-inal problems is obtained. Then, the multigrid algorithm based on the NBR is given.The convergence and the rate of convergence of the algorithm is analyzed. Besides,the error estimates of the algorithm are obtained. Finally, some numerical examples are presented to illustrate the feasibility and efficiency of the method.