In this work we analyze several possibilities to approximate the linear wave equation. We consider both the irreducible and the mixed formulation and, in each case, the possibility of working in the physical time or in the frequency domain. In the case of mixed formulations, we also discuss three possible functional frameworks in which the problem can be written, depending on the regularity chosen for the unknowns. The approximation to all problems posed can be done under the umbrella of the Variational Multiscale (VMS) method, which yields stable and accurate solutions in all cases. Of particular interest is the case of mixed methods, in which it is shown that the choice of the stabilization parameters allows one to switch from one functional setting to another.

Approximating the wave equation using VMS: irreducible vs mixed, time vs frequency and the choice of the functional setting

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