基于多极边界元法的三维位势问题解的存在唯一性

Existence and Uniqueness Property of the Three-dimensional for Potential Problems Based on Fast Multipole-BEM

目的多极边界元法是一种新型的边界元数值算法,已经建立了多极展开法和多极边界元法的基本理论知识,为多极展开法融合于边界元法格式和后面要研究的多极边界元法奠定了理论基础,给出了多极展开法适用的范围,本法更新了传统边界元法理论及计算结构,适应大规模运算工程问题的需要,因此需要研究基于多极边界元法的三维位势问题的解的存在唯一性.方法用到的方法有Krylov子空间方法的基础理论,GMRES算法,基于多极展开法的广义极小残余算法,多极展开法,三维位势问题多极边界元法.结果得到了三维位势问题多极边界元法解的存在唯一性及其证明,完善了三维位势问题多极边界元法的数学理论.结论建立了三维位势问题多极边界元法的初步理论,完善了三维位势问题多极边界元法的数学方法.

Objective The Fast Multipole-boundary Element Method （FM-BEM） is the new algorithm of the Boundary Element Method （BEM）. The basic theory knowledge of multipole expansion and the fast multipole boundary element method have been established, as well as the foundation of the multipole expansion method fusing with BEM form and the following to study multipole expansion BEM.With the discussion of the multipole expansion method＇s suitable scope, this method renews the traditional BEM theory and calculation structure, which is suitable for the problem of big scale operate project. Therefore, it is necessary to study existence and uniqueness property of the three-dimensional for potential problems based on Fast Multipole-Boundary Element Method. Methods The basic theory of Krylov Subspace Method; GMRES algorithm; the GMRES algorithm based on the Fast Multipole Method （FMM）; FMM; the three-dimensional for potential problems based on Fast Multipole-BEM. Results The existence and uniqueness property of the three-dimensional Fast Multipole BEM for potential problems are obtained, and the relevant proof is given, and the mathematical theory of the FM-BEM for 3D potential problems can be completed. Conclusion The three-dimensional Fast Multipole BEM for potential problems is established primarily; the mathematical method of the three-dimensional FM-BEM is perfected.