When you have fine-scale features (e.g. boundary layers) in the solution, its FEM approximation on coarse meshes converge at strange apparent rates. Looking at Cea's lemma, is this behaviour because of:

  • the best approximation error behaving strangely?
  • or the constant changing with h?
  • or should I actually look at Strang's first lemma? (errors in numerical integration of very fine-scale data)
  • $\begingroup$ This is partially speculation, so adding this as a comment rather than an answer: From some intuition and experience, I've generally attributed this to the boundary conditions themselves. Most lemmas/error/convergence estimates have an underlying assumption of periodicity -- various boundary conditions in practice can severely alter convergence rates. An example related to boundary layers I see regularly is pressure waves emanating from a body and getting partially reflected by an imperfect non-reflecting farfield BC. When those non-physical features die out, the convergence improves rapidly. $\endgroup$ – Aurelius Oct 15 '13 at 14:44
  • $\begingroup$ @Aurelius I think you can also observe the same phenomenon in a simple Poisson problem with homogeneous Dirichlet conditions when you have big oscillations in the rhs data. $\endgroup$ – gokturk Oct 16 '13 at 6:23

It is because we typically neglect higher order terms in error estimates. For example, we can show that $$ \|e\| \le C(u) h^2 + {\cal O}(h^3). $$ The point is that when $h$ is small, the cubic term is small and can be neglected. In fact, when $h$ is small, you can observe quadratic convergence. But whenever $h$ is not small (where "small" is relative to the features of the solution), the cubic and possibly even higher order terms can be large compared to $Ch^2$ and will lead to behavior that is not as easily predictable.

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    $\begingroup$ Thanks for the answer, but it's not precisely the answer to my question. My question would be, where do the higher order terms originate? From the h-dependency of the constant in Céa's lemma, or from the best approximation error? (This question was in fact inspired by your answer to an earlier question) $\endgroup$ – gokturk Oct 16 '13 at 6:16
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    $\begingroup$ If you consider the Laplace equation, the finite element solution is in fact the best approximation, not just within a constant factor of the best approximation. However, in the interpolation estimate, you end up with higher order terms (essentially, the higher order terms you get when you look at the Taylor series remainder term). $\endgroup$ – Wolfgang Bangerth Oct 16 '13 at 10:45

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