The governing equation
\begin{equation*}
\mathbf{M}\ddot{\mathbf{u}} +\mathbf{C}\dot{\mathbf{u}}+\mathbf{K}\mathbf{u}=\mathbf{f}
\end{equation*}
is linear assuming that $\mathbf{M}$, $\mathbf{C}$, $\mathbf{K}$ are constant.
However the equations governing an elasticity problem are linear only under the assumption of small displacements and small strains, which makes them absolutely inadequate for computer graphics applications. (Strange things occur if one uses equations valid at the small displacement limit, when finite displacements are analysed, like objects changing their size when subjected to rigid rotations.)
This said I think that what you are interested in is actually a Model Reduction technique, i.e. given a fine mesh which gives accurate results for your problem, to define an equivalent reduced model, which has much less degrees of freedom but still correctly describes the dynamics of the system under study. Typically this is done not by solving on a coarse mesh, but by suitably defining a projection on a lower dimensional space with the desired properties.
Literature is vast on the argument, but a quick check on google let me find a link to this ACM SIGGRAPH 2012 Course:
FEM Simulation of 3D Deformable Solids: A practitioner's guide to
theory, discretization and model reduction
please see http://www.femdefo.org
EDIT to address a question in a comment.
As pointed out by Wolfgang in his answer, a frame-by-frame comparison between two different meshes does not make much sense: $u_H \neq I_H u_h$.
Model reduction tries to project some of the information gained from the solution on the fine mesh onto a coarser mesh to speed up dynamic computations. But of course there is no silver bullet here, and every speed-up comes at the cost of some inaccuracy on the reduced model. Think it as the spectrum of $\mathbf{M}^{-1} \mathbf{K}$: if you do model reduction correctly, then in a given frequency range this matrix will have almost the same eigenvalues/eigenvectors for both the fine mesh and the reduced model. But of course the complete model has much more eigenvalues/eigenvectors than the reduced one, so its dynamics has to be different from the one of the reduced one.