Applying the parallel multistep methods presented in the talk by Lehel Banjai to the wave equation leads to a decoupled system of Helmholtz problems with complex wave numbers. In this talk we report on the solution of these decoupled Helmholtz problems using finite element methods. In particular we focus on the iterative solution of the resulting linear systems. Due to the indefinite nature of the discrete Helmholtz systems the efficient solution is not straight forward. We use multigrid-based preconditioning to define fast solvers. The overall complexity of the method is optimal with respect to the dimension of the space discretization. The dependence of the complexity on the number of time steps to be evaluated in parallel (number of Helmholtz problems to be solved) is less clear. The number of time steps affects the wavenumber distribution and therefore the performance of the linear solver. In the current setting there is a lack of efficiency for very large numbers of parallel time steps. We present a restarting technique to circumvent this drawback and to balance the advantages and disadvantages of parallel and sequential time discretization approaches. The presentation is supplemented by several numerical experiments.
This is joint work with Lehel Banjai.