Conventional methods to solve the time-harmonic elastic wave equations usually rely on either direct solvers or iterative solvers. The former are very efficient for treating multiple right hand side problems, as the matrix factorization needs to be done only once for all the right hand sides. However, it suffers from a significant shortcoming associated with high memory consumption and lack of scalability. The latter are matrix-free, and therefore much lighter in memory and scalable. However, dedicated preconditioners are required to converge these methods. The efficiency of existing preconditioners quickly deteriorates as the frequency increases. Another approach to compute time-harmonic solution to elastic wave equations is to consider timedomain solvers. Instead of computing the stationary solution, which convergence is shown to be dependent on the presence of trapped waves and complex wave phenomenon, we develop here a numerical strategy based on a controllability method. The method has been recently analyzed in the frame of acoustic propagation and we extend it here in the frame of linear elasticity. We rely on a spectral element space discretization and a fourth order Runge Kutta time integration. We present the basic properties and formulation of the method, before investigating its scalability and its memory requirement on canonical three-dimensional numerical experiments. The method is shown to be scalable for a problem involving approximately 250 millions degrees of freedom up to more than fifteen hundred computational units.