When solving the linear space-time wave equation in the band limited case, it is attractive to reduce the problem to one in space only by applying the Fourier transform in time. The result is the Helmholtz equation, a second order PDE in space which is not coercive in standard settings. Its discretization yields large frequency-dependent non-normal complex linear systems which are notoriously hard to solve, especially in the high frequency case. Because of the size of the systems, iterative methods are often required.
In the talk we describe a new multilevel domain decomposition method for efficiently solving these systems iteratively and we outline its convergence theory. This involves:
- the analysis of nearby problems with artificial absorption;
- a non-standard projection-theoretic setting for domain decomposition and
- the ``field of values'' analysis of the convergence of Krylov iterative methods.
All theory known at the present assumes the wave speed is constant. We also describe some recent progress in the removal of this assumption.
The talk is joint work with Euan Spence, Eero Vainikko and Stefan Sauter.