三维位势问题快速多极算法整体误差的收敛定理

CONVERGENCE THEOREM OF THE GLOBAL ERROR OF THE FAST MULTIPOLE METHOD FOR POTENTIAL PROBLEMS IN 3-D

快速多极算法是加速计算由许多物理问题得出的大型稠密线性方程组的一种有效算法.本文研究了求解三维位势问题快速多极算法整体误差的收敛性问题.首先推导了整体误差的表达式,然后给出了误差上界.其次将结果应用于自适应八叉树结构,得到具体的误差收敛阶.最后通过具体的数值算例验证了本文的结果.本文的方法和结论也可以推广到计算弹性静力学问题和斯托克斯流问题的快速多极算法的误差分析中.

The fast multipole method(FMM)can accelerate the iterative solver of the large dense linear equations arising from many physical problems.This article is concerned with the convergence of the FMM for three dimensional potential problems.Firstly,derive the expression of the global error,and then give a novel estimate of the error bound.Secondly,the result is applied to the adaptive octree structure,and the specific convergence order is obtained.Finally,an illustrative example is provided to test the proposed results.The method of this paper can also be used to estimate the error of the FMM for elastostatic problems and Stokes flow problems.