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Is there any mesh size restriction of spatial discretization in FEMs and FDMs (finite difference)?

If a mesh is very coarse there is still perhaps nothing to stop the program from running and cause reporting errors, making me conclude that there is no such requirement on the spatial mesh size as that on the time step length that is restricted in time integration for its stability. If not, is there any counterexample?

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    $\begingroup$ If the discretisation fits your boundary conditions very poorly you may run into issues. Additionally if an FEM mesh has very elongated triangles that can lead to numerical issues. There are also opposite examples, where if your discretisation is too fine your condition number may be too large, and if you're discretising a singular problem then the finer the discretisation the worse things get. $\endgroup$
    – lightxbulb
    Feb 10 at 9:14
  • $\begingroup$ How do you define "the program runs successfully"? In other word, how exactly do you decide whether a restriction is satisfied or not? $\endgroup$ Feb 10 at 18:00
  • $\begingroup$ @WolfgangBangerth no sign of divergence or unwanted wiggles $\endgroup$
    – feynman
    Feb 11 at 3:09
  • $\begingroup$ @lightxbulb thanks very much. Then I'll make sure the BCs match well the ICs and if the mesh isn't so coarse as to distort the elements shapes too much. $\endgroup$
    – feynman
    Feb 11 at 3:12
  • $\begingroup$ @feynman It is very difficult to construct schemes that have no unwanted wiggles at all. If the wiggles' amplitude is less than $10^{-3}$ is that good enough? If they are larger, is that failure? $\endgroup$ Feb 11 at 19:38

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Keep in mind that just because the simulation doesn't report any errors, doesn't mean you got the correct solution to the problem you are solving. Sometimes, the reason for that deviation might just be a spatial step that is not small enough.

If the spatial step is too big, you might not correctly catch your solution's gradient. The deviation of the solution you got might be unacceptable. The way to check this is usually easy. Once you have done the simulation using a coarse spatial mesh, do it once more with a finer spatial mesh. If the two solutions don't nearly match and both of the solutions are stable, then the spatial step was too big in the first simulation.

Also, depending on the exact problem you are solving, there may be spatial step restrictions. For example, if you are solving a hyperbolic PDE, with initial or boundary conditions that contain high-frequency modes, the solution might contain dispersion which causes the simulation to be useless. Here is an example of such a problem where both the spatial and time steps were restricted in order to get a usable solution. Namely, the group velocity had to be equal to phase velocity which restricted the CFL number to the value of one, allowing one to derive the exact restriction formula for the step size. In fact, it gets worse if you are solving a PDE with two spatial dimensions because of numerical anisotropy. In that case, not only the spatial step needs to be chosen carefully, but the discretization scheme as well.

This shows that restrictions to spatial step size can definitely exist. It should be chosen to fit the problem you are solving.

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