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

Abstract. We present a solver for the 2D high-frequency Helmholtz equation in heterogeneous, constant density, acoustic media, with online parallel complexity that scales empirically as O(NP ), where N is the number of volume unknowns, and P is the number of processors, as long as P = O(N1/5). This sublinear scaling is achieved by domain decomposition, not distributed linear algebra, and improves on the P = O(N1/8) scaling reported earlier in [44]. 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, e ciently solved via a nested version of the polarized traces preconditioner [44]. The favorable complexity is achieved via an e cient application of the integral operators involved in the SIE.