I use FEM to simulate deformations of elastic objects for animation applications in computer graphics. The governing equation is generally with the form:$$\mathbf{M}\ddot{\mathbf{u}} +\mathbf{C}\dot{\mathbf{u}}+\mathbf{K}\mathbf{u}=\mathbf{f}$$where $\mathbf{M}$ is the mass matrix, $\mathbf{C}$ is the damping matrix, $\mathbf{K}$ is the stiffness matrix, and $\mathbf{u}$ is the displacement to be solved for.I'm trying to solve this time-dependent equation using FEM with very coarse spatial resolutions, mainly to save computational time. As far as I know, the FEM solution converges to the exact solution as spatial resolution goes to infinity. What I'm trying to figure out is: what is the relationship between FEM solutions using different spatial resolutions besides that they converge to the same exact solution? What I'm hoping to achieve is to get coarse solution as the downsampling of fine solution, since fine solution is closer to exact solution. Is this even possible? 
Relationship between FEM solutions of PDE with different spatial resolutions
pde;finite element;simulation
It doesn't generally work this way. You want to compare $u_H$ and $u_h$ computed on coarse and fine meshes $T_H$ and $T_h$. It is generally not true that $u_H=I_H u_h$ where $I_H$ is the interpolation to the coarse mesh. Nor is it generally true that $u_H=P_H u_h$ for some projection operator (though that is indeed the case for elliptic problems if you could integrate coefficients exactly), but even if it were, it would not yield an easily understandable relationship. In general, you don't even know that $u_h$ is more accurate than $u_H$ (in the sense that $\|u-u_h\| < \|u-u_H\|$) on coarse meshes, as convergence only exists in the limit and not for a particular choice of $h,H$.So, in essence, there is not very much you can say about the solutions on different meshes.