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The simulation of the elastodynamics equations at high-frequency suffers fromthe well known pollution effect. We present a Petrov--Galerkin multiscalesub-grid correction method that remains pollution free in natural resolutionand oversampling regimes. This is accomplished by generating corrections tocoarse-grid spaces with supports determined by oversampling lengths related tothe $\log(k)$, $k$ being the wavenumber. Key to this method arepolynomial-in-$k$ bounds for stability constants and related inf-sup constants.To this end, we establish polynomial-in-$k$ bounds for the elastodynamicsstability constants in general Lipschitz domains with radiation boundaryconditions in $\mathbb{R}^3$. Previous methods relied on variationaltechniques, Rellich identities, and geometric constraints. In the context ofelastodynamics, these suffer from the need to hypothesize a Korn's inequalityon the boundary. The methods in this work are based on boundary integraloperators and estimation of Green's function's derivatives dependence on $k$and do not require this extra hypothesis. We also implemented numericalexamples in two and three dimensions to show the method eliminates pollution inthe natural resolution and oversampling regimes, as well as performs well whencompared to standard Lagrange finite elements.

Multiscale Sub-grid Correction Method for Time-Harmonic High-Frequency Elastodynamics with Wavenumber Explicit Bounds