The Research Of Fast Multipole Boundary Element Method For Boundary Value Problems Of Two Classes Of Elliptic Partial Differential Equation

Fast Multipole Boundary Element Method (FM-BEM) is a newly developednumerical algorithm, which combines the Fast Multipole Method (FMM) with theBoundary Element Method (BEM). The operations and memory requirement areproportional to the unknowns N. Combined with the FMM, the BEM can be used to solvelarge-scale problems with hundred millions of DOF on one computer within several hours,which provides widespread development space for the BEM. Compared with theconventional BEM, the FM-BEM can greatly reduce the computational complexity andmemory requirement. As a highly efficient method, the FM-BEM is suitable forlarge-scale numerical computation. The purpose of this paper is to present the FM-BEMfor the solution of the boundary value problems of elliptic partial differential equation, andto analyze the truncation error in theory and in practical computations, through which thepresented method can be proved to be accurate and efficient. Then a completefundamental theory system was tried to establish for the FM-BEM, which was suitable forthe large-scale numerical computing.The paper was divided into five chapters. In the first chapter, for the BEM and theFMM, the development history, research progress and present state of were overviewed.The advantages of the BEM and the basic idea of the FMM were analyzed. The meaning,research progress and the basic idea of the FM-BEM were discussed. Then itsachievements of later years were summarized. The second chapter was on the FM-BEMfor solving two dimensional (2-D) potential problems. Detailed derivations of theformulations, discussions on the algorithms for the FM-BEM are presented for2-Dpotential problems, which will serve as the prototype of the FM-BEM for all otherproblems discussed in the subsequent chapters. The third chapter presented a kind ofFM-BEM based on Legendre series for the solution of3-D potential problems. The basicFM-BEM formulations and the main steps of the FM-BEM were introduced. Then, thetruncation errors of multipole expansion and local expansion were discussed.The approaches and results developed in Chapters2and3are extended in the following two chapters to solve2-D (Chapter4) and3-D (Chapter5) Helmholtz problems.In each case, the related BIE formulations are presented first, and the same systematicFM-BEM approaches developed for2-D and3-D potential problems are extended to therelated FM-BEM formulations for the subject of chapter4and chapter5. In chapter4andchapter5, the multipole expansion theorems for the fundamental solution of2-D and3-DHelmholtz problems were given, which were the most important thing for the FM-BEMbased on series form expansion. Then, the truncation error was analyzed and proved to becontrolled by a truncation number p. The different approximate expressions of p for2-D and3-D Helmholtz problems were deduced and proved to be crucial to thecomputational precision, respectively.