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

Abstract

A new, coercive formulation of the Helmholtz equation was introduced in [1]. 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.

Wavenumber-explicit analysis for the Helmholtz h-BEM: error estimates and iteration counts for the Dirichlet problem

“…4. The use of identities originally due to Morawetz [66] to prove coercivity of A k,η [75] and to introduce new coercive formulations of Helmholtz problems [74,65,42,43,28].…”

A sharp relative-error bound for the Helmholtz h-FEM at high frequency

“…4. The use of identities introduced in [86] to prove coercivity of boundary-integral operators [94] and to introduce new coercive formulations of Helmholtz problems [30,55,56,85,93]. 5.…”

Convergence of classical optimized non-overlapping Schwarz method for Helmholtz problems in closed domains

“…In order to find the best symbol λ, we need to determine the convergence radius of the iterative scheme (5). This objective can be achieved by: 1. deriving the fundamental solutions of (5a) and (5d); 2. fixing the integration constants with the boundary conditions (5c) and (5f) and the definition (6); 3. determining the solutions of (5a) and (5d) from the expressions found in steps 1 and 2; 4. computing ∂ p n i (x, s) ∂x at x = 0 from the solutions p n i (x, s) found in step 3;…”

“…It is well known that large-scale time-harmonic Helmholtz problems are hard to solve because of i) the pollution effect [1] and ii) the indefiniteness of the discretized operator [2]. While the pollution effect can be alleviated by using higher order discretization schemes [3], the indefiniteness is an intrinsic property of time-harmonic wave problems, at least with standard variational formulations [4,5], which significantly limits the performance of classical iterative solvers, such as the generalized minimal residual method (GMRES). Of course, as an alternative to iterative algorithms, direct solvers can be used.…”