## Section: New Results

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

This work studies preconditioning the Helmholtz and Maxwell equations, where the preconditioner is constructed using two-level overlapping Additive Schwarz Domain Decomposition. 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. An important feature of the theory is that it allows the coarse space to be built from low-order elements even if the PDE is discretised using high-order elements. This theory is presented in [24] and is illustrated by numerical experiments, which also (i) explore replacing the PEC boundary conditions on the subdomains by impedance boundary conditions, and (ii) show that the preconditioner for the problem with absorption is also an effective preconditioner for the problem with no absorption. The numerical results include two substantial examples arising from applications; the first (a problem arising in medical imaging from the Medimax ANR project) shows the robustness of the preconditioner against heterogeneity, and the second (scattering by a COBRA cavity) shows good scalability of the preconditioner with up to 3000 processors. The parallel implementation was done using FreeFem++ and HPDDM. We performed additional numerical studies of this two-level Domain Decomposition preconditioner for the Maxwell equations in [23], and for the Helmholtz equation (in 2D and 3D) in [25], where we also compare it to another two-level Domain Decomposition preconditioner where the coarse space is built by solving local eigenproblems on the interface between subdomains involving the Dirichlet-to-Neumann (DtN) operator.