Relationship between FEM solutions of PDE with different spatial resolutions

Question

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?

Answer 1

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.

Answer 2

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.