A Robust Multilevel Method for Hybridizable Discontinuous Galerkin Method for the Helmholtz Equation

TL;DR

This paper introduces a robust multilevel preconditioner for the hybridizable discontinuous Galerkin method applied to the Helmholtz equation with high wave numbers, improving convergence and efficiency.

Contribution

It proposes a novel multilevel preconditioner with optimized intergrid transfer and GMRES smoothing, enhancing solver robustness for high-frequency Helmholtz problems.

Findings

Convergence is mesh independent for fixed wave numbers.

Algorithm performance is mildly affected by wave number.

Numerical results confirm improved efficiency and robustness.

Abstract

A robust multilevel preconditioner based on the hybridizable discontinuous Galerkin method for the Helmholtz equation with high wave number is presented in this paper. There are two keys in our algorithm, one is how to choose a suitable intergrid transfer operator, and the other is using GMRES smoothing on coarse grids. The multilevel method is performed as a preconditioner in the outer GMRES iteration. To give a quantitative insight of our algorithm, we use local Fourier analysis to analyze the convergence property of the proposed multilevel method. Numerical results show that for fixed wave number, the convergence of the algorithm is mesh independent. Moreover, the performance of the algorithm depends relatively mildly on wave number.