Parallel controllability methods for the Helmholtz equation

Abstract

The Helmholtz equation is notoriously difficult to solve with standard numerical methods, increasingly so, in fact, at higher frequencies. Controllability methods instead transform the problem back to the time-domain, where they seek the time-harmonic solution of the corresponding time-dependent wave equation. Two different approaches are considered here based either on the first or second-order formulation of the wave equation. Both are extended to general boundary-value problems governed by the Helmholtz equation and lead to robust and inherently parallel algorithms. Numerical results illustrate the accuracy, convergence and strong scalability of controllability methods for the solution of high frequency Helmholtz equations with up to a billion unknowns on massively parallel architectures.

Solving frequency-domain elastic wave equations via parallel controllability methods

“…In other words, letting the wave equation propagates in a long-term yields the solution of (1) (Nihei and Li, 2007). However, in the presence of physical boundary conditions on the free surface the wave may be trapped or contain resonance such that the convergence may deteriorate and be slow -see Grote and Tang (2019); Grote et al (2020).…”

“…Another existing approach relies on the computation of frequencydomain solutions from time-domain solvers (Nihei and Li, 2007;Sirgue et al, 2008), which takes benefit on the reliability and good scaling properties of such solvers. Related to such last methods, the controllability approach has been shown to be very efficient and robust for solving Helmholtz equations (Bristeau et al, 1998;Mönkölä, 2010;Grote and Tang, 2019;Grote et al, 2020). The controllability method (CM) requires repeated solutions of the time-domain wave equation in a short time-window.…”

An adaptive finite element method for high-frequency scattering problems with smoothly varying coefficients

“…Let u be the solution to the wave equation given in (25), with f (x, t) := F (x)e −iωt for all t > T for some T > t 0 , and let U be the solution to the Helmholtz equation given in (24). The limiting amplitude principle states that (26) lim…”

“…• For d = 3, (26) was derived in [41] for (26) of the limiting amplitude principle is not valid [13,Sect. 3]; a modified form is currently under investigation and will be presented in a forthcoming paper.…”

“…Classical time domain methods are the finite difference and finite element time domain methods [44,40]. Time-domain methods that are specifically devised for solving frequency-domain problems include the controllability method [11,23], with its spectral version [29] and its extensions [28,26], the WaveHoltz method [4], and the time-domain preconditioner of [39].…”