# What is the origin of the preasymptotic convergence behaviour in FEM?

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)
• 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. Oct 15 '13 at 14:44
• @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. 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.