Nested Domain Decomposition with Polarized Traces for the 2D Helmholtz Equation

Abstract

We present a solver for the two-dimensional high-frequency Helmholtz equation in heterogeneous, constant density, acoustic media, with online parallel complexity that scales empirically as ${\cal O}(\frac{N}{P})$, where $N$ is the number of volume unknowns, and $P$ is the number of processors, as long as $P = {\cal O}(N^{1/5})$. This sublinear scaling is achieved by domain decomposition, not distributed linear algebra, and improves on the $P = {\cal O}(N^{1/8})$ scaling reported earlier in [L. Zepeda-Nún͂ez and L. Demanet, J. Comput. Phys., 308 (2016), pp. 347--388]. The solver relies on a two-level nested domain decomposition: a layered partition on the outer level and a further decomposition of each layer in cells at the inner level. The Helmholtz equation is reduced to a surface integral equation (SIE) posed at the interfaces between layers, efficiently solved via a nested version of the polarized traces preconditioner [L. Zepeda-Nún͂ez and L. Demanet, J. Comput. Phys., 308 (2016), pp. 347--388]. The favorable complexity is achieved via an efficient application of the integral operators involved in the SIE.