Can coercive formulations lead to fast and accurate solution of the Helmholtz equation?
For researchers solving Helmholtz problems, this work offers theoretical guarantees for iterative solver performance, but the practical benefit is limited as iteration counts still grow with k.
The paper analyzes a coercive formulation of the Helmholtz equation, showing it suffers from the same pollution effect as standard formulations, but provides the first rigorous bounds on GMRES iterations for a preconditioned Helmholtz system with a symmetric positive-definite preconditioner, though iteration count grows with wavenumber k.
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.