Helmholtz DD preconditioners

DG methods for Helmholtz can naturally encorporate Robin-type transmisson conditions. This is a key ingredient here, that allows us to build a domain decomposition preconditioner by restricting the global matrix.

METIS subdomains and the sorted TDG matrix

Split the mesh into subdomains, get the (Trefftz)-DG dofs of the subdomains, then sweep through the local blocks. CG then uses these multiplicative Schwarz sweeps as a preconditioner, so each iteration combines local subdomain corrections with the global Krylov iteration.

CG + multiplicative Schwarz convergence on a trapping geometry

Nice convergence and a remarkable, sudden appearance of the trapped mode. After a few iterations where the wave field still behaves stubbornly, the residual drops rapidly.

Submarine scattering

The same idea also scales to a larger scattering example around a submarine, with almost 10 million DoFs split over 100 subdomains.