In this talk we will present recent improvements to the quasi-optimal domain decomposition method for the Helmholtz equation presented in ^{1}. The key point of the method is the construction of an accurate local approximation of the exact Dirichlet-to-Neumann operator which leads to a new transmission operator between sub-domains. We will show that this local approximation, based on complex Pad approximants, is well-suited for large scale parallel finite element simulations of high frequency scattering problems, with either manual or automatic mesh partitioning. In particular, we will show that our algorithm is quasi-optimal in the sense that the convergence rate of the iterative solver depends only slightly on both the frequency and the mesh refinement.
Y. Boubendir, X. Antoine and C. Geuzaine, A Quasi-Optimal Non-Overlapping Domain Decomposition Algorithm for the Helmholtz Equation. Journal of Computational Physics 231 (2), (2012), pp.262-280 ↩︎