## Section: New Results

### Domain decomposition preconditioning for high frequency wave propagation problems

The work about domain decomposition preconditioning for Maxwell equations has been published in [21]. It studies two-level preconditioners where the coarse space is based on the discretisation of the PDE on a coarse mesh. The PDE is discretised using finite-element methods of fixed, arbitrary order. The theoretical part of this work is the Maxwell analogue of a previous work for Helmholtz equation, and shows that for Maxwell problems with absorption, if the absorption is large enough and if the subdomain and coarse mesh diameters are chosen appropriately, then classical two-level overlapping Additive Schwarz Domain Decomposition preconditioning performs optimally – in the sense that GMRES converges in a wavenumber-independent number of iterations. The theory is also illustrated by various numerical experiments.

Ongoing studies are being conducted on recursive one-level optimized Schwarz methods for the high frequency Helmholtz and Maxwell equations. The method consists in solving the subdmain problems in a one-level optimized Schwarz preconditioner only approximately, using inner GMRES iterations preconditioned again by a one-level method, with smaller subdomains. The asymptotic behaviour and parallel scalability of the method are being investigated. Exhaustive numerical experiments are being conducted to compare the efficiency of this method with two-level preconditioners, including cavity problems and benchmarks in seismic imaging.