Nested domain decomposition with polarized traces for the 2D Helmholtz equation

TL;DR

This paper introduces a scalable parallel solver for the 2D high-frequency Helmholtz equation using nested domain decomposition and polarized traces, achieving near-linear complexity with respect to the number of unknowns.

Contribution

It presents a novel nested domain decomposition method with polarized traces for efficient parallel solution of the 2D Helmholtz equation, improving scalability over previous approaches.

Findings

Empirical parallel complexity scales as O(N/P) with P=O(N^{1/5})

The method reduces the Helmholtz problem to a surface integral equation at interfaces

Achieves sublinear scaling through efficient integral operator application

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 $\mathcal{O}(\frac{N}{P})$, where $N$ is the number of volume unknowns, and $P$ is the number of processors, as long as $P = \mathcal{O}(N^{1/5})$. This sublinear scaling is achieved by domain decomposition, not distributed linear algebra, and improves on the $P =\mathcal{O}(N^{1/8})$ scaling reported earlier in [L. Zepeda-N\'u\~nez 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-N\'u\~nez 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.