Multilevel Preconditioner with Stable Coarse Grid Corrections for the Helmholtz Equation

TL;DR

This paper introduces a robust multilevel preconditioner for the Helmholtz equation that uses CIP-FEM for stable coarse grid corrections, demonstrating efficiency and mild wave number dependence in numerical tests.

Contribution

It presents a novel multilevel preconditioner utilizing CIP-FEM for stable coarse grid corrections, improving convergence for high wave number Helmholtz problems.

Findings

Efficient for a reasonable range of frequencies

Performance depends mildly on wave number

Only one GMRES smoothing step can ensure optimal convergence

Abstract

In this paper we consider a class of robust multilevel precontioners for the Helmholtz equation with high wave number. The key idea in this work is to use the continuous interior penalty finite element methods (CIP-FEM) studied in \cite{Wu12,Wu12-hp} to construct the stable coarse grid correction problems. The multilevel methods, based on GMRES smoothing on coarse grids, are then served as a preconditioner in the outer GMRES iteration. In the one dimensional case, convergence property of the modified multilevel methods is analyzed by the local Fourier analysis. From our numerical results, we find that the proposed methods are efficient for a reasonable range of frequencies. The performance of the algorithms depends relatively mildly on wave number. In particular, only one GMRES smoothing step may guarantee the optimal convergence of our multilevel algorithm, which remedies the shortcoming of the multilevel algorithm in \cite{EEO01}.