The Helmholtz equations are the frequency-domain wave equations. Both the acoustic and the elastic versions of this equation are challenging numerically: the resulting linear system is indefinite, which leads to failure of many iterative solvers due to the oscillatory nature of the solutions. In some applications, such as inverse problems and underground imaging, the elastic version is needed. The elastic Helmholtz operator is known as harder to solve and fewer solvers address it. We show that, in fact, the difficulty in solving the elastic Helmholtz equation stems from the acoustic one. We develop a reduction-based approximate-commutator preconditioner for the elastic Helmholtz equation. By utilizing the commutation relations of the underlying continuous differential operators, we suggest a block-triangular preconditioner whose diagonal blocks are acoustic operators. Thus, we enable the solution of the elastic version using virtually any existing solver for the acoustic version. We prove a sufficient condition for the convergence of our method, exploring the spectral behavior of the underlying operators.  We show that the resulting reduction achieves wavenumber independent convergence. However, solving the acoustic Helmholtz equation efficiently is necessary for efficient application of this reduction. Standard multigrid methods fail to converge for high frequency Helmholtz problems, and the performance of related preconditioners strongly depends on the wavenumber. Using novel symbol analysis aimed at minimizing numerical dispersion, we develop a multigrid method for the acoustic Helmholtz equation, which achieves wavenumber independent convergence, even in the high-frequency regime. Overall, our methods form fast and scalable solvers for high-frequency waves in highly heterogeneous media in 2D and in 3D. The results presented in this lecture were obtained during the phd thesis of the speaker, under the supervision of Eran Treister