A Spectral Method for Integral Formulations of Potential and High-Frequency Scattering Problems
Johannes Tausch (Southern Methodist University)
We discuss a spectral method to reduce the computational complexity
associated with boundary integral operators. In this approach the
Green's function is replaced by a trigonometric expansion, which is
valid globally for all positions of source- and field points. The
resulting integral operator can be evaluated efficiently using
non-equispaced FFTs. Since in the case of a singular Green's function
the convergence rate of the Fourier series is slow, the method is
applied to a mollified version of the Green's function. The remainder
can be expanded with respect to the mollification parameter and leads
to a diagonal operator.
For the integral equations associated with the Laplace equation the
mollification parameter and the number of terms in the Fourier series
can be chosen such that the scheme is nearly asymptotically optimal.
The talk will present some comparisons of this approach and the Fast
Multipole Method. Moreover, we will present some applications to high
frequency scattering problems.