Domain Decomposition methods for the high frequency Helmholtz equation

Abstract: 

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:

1. the analysis of nearby problems with artificial absorption;
2. a non-standard projection-theoretic setting for domain decomposition and
3. 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.