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

• Published in: Journal of computational and applied mathematics Vol. 352; pp. 110 - 131
• Main Authors: Diwan, Ganesh C., Moiola, Andrea, Spence, Euan A.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 15.05.2019
• Subjects: Coercive variational formulation; Finite element method; GMRES; Helmholtz equation; Pollution effect; Wavenumber-explicit analysis; Finite element method; 65F10; GMRES; Helmholtz equation; Wavenumber-explicit analysis; Coercive variational formulation; 35J05; Pollution effect; 65N30
• ISSN: 0377-0427; 1879-1778
• DOI: 10.1016/j.cam.2018.11.035

A new, coercive formulation of the Helmholtz equation was introduced in Moiola and Spence (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→∞, 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.