The Improvement Solver Of FM-BEM And Error Analysis Of Legendre Series Fundamental Solutions

Fast Multipole Boundary Element Method(FM-BEM)is a combination of the fast multipole expansion method(FMM)and the boundary element method(BEM).FMM method is an approximate calculation method.It is based on spherical harmonics of multipole expansion in the space and can reduce the order of magnitude to O(N)by using the recursive algorithm structure.In the realization of BEM,a key factor is the solution to the final linear equations.Krylov subspace generalized minimal residual algorithm(GMRES(m))is a more effective solver.In the implementation of GMRES(m)algorithm,once the restart parameter m is chosen,it will keep fixed in the whole iteration process.So the selection of parameters m is also one of the main factors which influences the effective implementation of the algorithm.Based on this,deep researches are made in this paper as follows:Firstly,the potential problems of Legendre series multipole BEM are studied.This dissertation presents an error analysis with regard to Legendre series multipole BEM for three-dimensional potential problems.The far-field partition criterion is present according to the requirement of precision.Then,the truncation errors are deduced to be concerned with truncated number and divide radius.The approximate expression of truncation error for potential problem is deduced and proved to be crucial to the practical calculation.Secondly,based on the GMRES(m)basic idea,a kind of GMRES(m)algorithm with variable restart parameters was proposed,namely,VRP-GMRES(m)algorithm.By properly changing the variable restart parameter for the GMRES(m)algorithm,the iteration stagnation problem resulted from improper selection of the parameter m is resolved efficiently.And the algorithm served as the fast solver for FM-BEM to solve three-dimensional elastic problems.The numerical examples show that VRP-GMRES(m)algorithm is feasible and effective.Finally,combining with Householder transform and variable restart parameter,a new iterative method named VHGMRES(m)algorithm is proposed.And the projection of the error vector m(10)1r on mr is used to prove that the proposed algorithm is not only fast convergent but also is highly accurate.And the conclusion was validated by numerical examples.