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Bound on global error of the fast multipole method for Helmholtz equation in 2-D
This paper analyze the global error of the fast multipole method(FMM) for
two-dimensional Helmholtz equation. We first propose the global error of the
FMM for the discretized boundary integral operator. The error is caused by
truncating Graf's addition theorem, according to the limiting forms of Bessel
and Neumann functions, we provide sharper and more precise estimates for the
truncations of Graf's addition theorem. Finally, using the estimates we derive
the explicit bound and convergence rate for the global error of the FMM for
Helmholtz equation, numerical experiments show that the results are valid. The
method in this paper can also be applied to the FMM for other problems such as
potential problems, elastostatic problems, Stokes flow problems and so on
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