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Solving time domain boundary integral formulations of the wave equation by a combination of FFT and panel-clustering techniques

## Stefan Sauter (University of Zürich)

In our talk we will present and analyse a new fast method for the numerical
solution of the time domain boundary integral formulations of the wave equation.
We employ Lubich's convolution quadrature method for the time discretization
and a Galerkin boundary element method for the spatial discretization.
The coefficient matrix of the arising system of linear equations is a triangular
block Toeplitz matrix. In the literature two approaches for its solution have
been proposed: (a) By using FFT techniques the computational complexity is
reduced substantially while the storage cost stays unchanged and is,
typically, very high. (b) By using panel-clustering techniques the gain is
reversed: the computational cost stays (approximately) unchanged while
the storage cost is reduced substantially.

In our talk, we will present a new fast method which combines the advantages
of the two approaches: First, the discrete convolution (related to the block
Toeplitz system) is transformed to the (discrete) Fourier image and, then,
a new panel-clustering method is applied to the transformed system. This
requires efficient (*H*- and *H*^{2}-matrix) representations
of integral operators related to Helmholtz-type equations for a large range of
complex wave numbers.

In our talk, we will focus on the methodolical approach and the convergence
analysis while the sparse approximations of the Helmholtz-type integral operators
for complex wave numbers will be presented by L. Banjai in the second
part of this talk.