Can coercive formulations lead to fast and accurate solution of the Helmholtz equation?

TL;DR

This paper investigates a new coercive formulation of the Helmholtz equation, analyzing its discretization and iterative solution, providing the first rigorous bounds on GMRES iterations with a positive-definite preconditioner.

Contribution

It offers the first rigorous bounds on GMRES iterations for a preconditioned coercive Helmholtz formulation, and compares its behavior to standard formulations.

Findings

Coercive formulation exhibits similar pollution effects as standard.

Derived k-explicit bounds on GMRES iterations with positive-definite preconditioning.

First rigorous analysis of iterative solver complexity for this formulation.

Abstract

A new, coercive formulation of the Helmholtz equation was introduced in [Moiola, Spence, SIAM Rev. 2014]. In this paper we investigate $h$-version Galerkin discretisations of this formulation, and the iterative solution of the resulting linear systems. We find that the coercive formulation behaves similarly to the standard formulation in terms of the pollution effect (i.e. to maintain accuracy as $k\to\infty$, $h$ must decrease with $k$ at the same rate as for the standard formulation). We prove $k$-explicit bounds on the number of GMRES iterations required to solve the linear system of the new formulation when it is preconditioned with a prescribed symmetric positive-definite matrix. Even though the number of iterations grows with $k$, these are the first such rigorous bounds on the number of GMRES iterations for a preconditioned formulation of the Helmholtz equation, where the preconditioner is a symmetric positive-definite matrix.