Multigrid-based ‘shifted-Laplacian’ preconditioning for the time-harmonic elastic wave equation

An MSSS-preconditioned matrix equation approach for the time-harmonic elastic wave equation at multiple frequencies

“…This particular choice of r * shows that the MSSS preconditioner has comparable performance to the multi-grid approaches in [24,34] where the authors numerically prove O(n 3 ) complexity for 2D problems of size n x = n y ≡ n.…”

“…In [1] a damped preconditioner for the elastic wave equation is presented. The authors of [34] analyze a multi-grid approach for the damped problem. Both works are extensions of the work of Erlangga et al [33] for the acoustic case.…”

Nested Domain Decomposition with Polarized Traces for the 2D Helmholtz Equation

“…The choice of the optimal complex-shift was studied by Cools and Vanroose [26] and by Gander et al [48]. Finally, the generalization of the complex shifted Laplacian preconditioner to the elastic wave equation has been studied by Rizzuti and Mulder [79]. We point out that most of the multigrid methods mentioned above exhibit a suboptimal dependence of the number of iterations to converge with respect to the frequency, making them ill-suited for high frequency problems.…”

An efficient two-level preconditioner for multi-frequency wave propagation problems

“…The focus of the present work, however, lies on situations where the discretization matrices K and M stem from a discretization of the time-harmonic elastic wave equation [13]. Depending on the specific choice of boundary conditions, the structure of the matrices varies, and the shifts s k are either equal to the (angular) wave frequencies [6,41] or to the squared (angular) wave frequencies [2,30]. For both situations, we will consider viscous damping by substituting s k → (1 − i)s k , where > 0 is the damping parameter and i ≡ √ −1, cf.…”