Scalable DPG Multigrid Solver for Helmholtz Problems: A Study on Convergence

TL;DR

This paper introduces a scalable multigrid preconditioner for high-frequency Helmholtz problems, demonstrating robust convergence and scalability to billion DOFs through advanced parallel implementation and novel coarse-grid strategies.

Contribution

It extends multigrid convergence results to larger problem sizes using a new coarse-grid approach and provides a detailed analysis of convergence behavior in high-frequency regimes.

Findings

Achieved convergence for systems with up to 10^9 DOFs.

Demonstrated h and p robustness in convergence.

Validated scalability on large seismic modeling benchmarks.

Abstract

This paper presents a scalable multigrid preconditioner targeting large-scale systems arising from discontinuous Petrov-Galerkin (DPG) discretizations of high-frequency wave operators. This work is built on previously developed multigrid preconditioning techniques of Petrides and Demkowicz (Comput. Math. Appl. 87 (2021) pp. 12-26) and extends the convergence results from $\mathcal{O}(10^7)$ degrees of freedom (DOFs) to $\mathcal{O}(10^9)$ DOFs using a new scalable parallel MPI/OpenMP implementation. Novel contributions of this paper include an alternative definition of coarse-grid systems based on restriction of fine-grid operators, yielding superior convergence results. In the uniform refinement setting, a detailed convergence study is provided, demonstrating h and p robust convergence and linear dependence with respect to the wave frequency. The paper concludes with numerical results on hp-adaptive simulations including a large-scale seismic modeling benchmark problem with high material contrast.