A Unbounded Domain Decomposition Method Based On The Natural Boundary Reduction

Many problems in science and engineering, such as physics, reactor, oil explorationand development and so on, can transform into the partial differential equations onunbounded domains by establishing models. There is a variety of methods for theproblems of bounded domain and the methods are relatively mature. But these methodsare not applicable to the problems of unbounded domain. Through long-termexploration, Chinese scholars first put forward domain decomposition algorithm basedon natural boundary reduction. It is a new kind of domain decomposition algorithm. It isquite effective for the unbounded domain problems and it has obvious advantages.This thesis first makes a brief introduction to the domain decomposition algorithm,and then introduces the natural boundary element method and its mathematical theory.Based on these, it discusses the unbounded domain decomposition algorithm based onnatural boundary reduction, including Schwarz alternating method and D-N alternatingmethod. By introducing artificial boundary, unbounded domain is separated into abounded domain and a typical unbounded domain. Then in bounded domain, we solvethe problem by combining with the artificial boundary conditions and using finiteelement method. But in the unbounded domain, we use the theory of natural boundarydomestication to solve it. This thesis introduces Schwarz alternating method and D-Nalternating method for a kind of unbounded domain. The author extends it to higherdimensional unbounded domains. By taking three-dimensional Helmholtz equationbased on the natural boundary reduction over a kind of unbounded space domain forexample, the paper gives the continuous and discrete D-N algorithm and discusses itsconvergence, also proves that the convergence speed is unrelated with mesh parameterh.