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In some of my computations I calculate a scalar value $\lambda_h$ (in my case an eigenvalue) depending on a finite element discretization of the domain. Usually we can manage to find estimates of the form $$ |\lambda - \lambda_h| \leq C h^k $$ where $k$ depends on the problem, the degree of the approximation, etc.

Question:

I was wondering under which circumstances (if any) it is possible to say that the error of a scalar function depending on the FE approximation has a Taylor-like expansion: $$ \lambda_h = \lambda_{exact} + C_1h^{\alpha_1}+C_2h^{\alpha_2}+...$$ where $\alpha_i$ is an increasing sequence.

If there are any relevant references, I am interested.


Context:

This is in connection with one eariler question of mine, regarding the good behavior of some extrapolation techniques applied to my problem. It turns out that the extrapolation I use (Wynn's epsilon algorithm) gives the limit of the sum of $n$ geometric sequences starting from $2n+1$ values. The existence of a Taylor-like expansion for the eigenvalue would justify this behavior since the first few Taylor coefficients will be cancelled when doing the extrapolation.

In the example presented below the initial computations using only finite elements give a convergence of order 2, while the extrapolation gives something like order 6, quickly running towards machine precision. See the picture: enter image description here

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  • $\begingroup$ Do you mean an expansion element wise, or globally? For the former case, and the less exotic element choices, your criterion is the same as asking if $\lambda_{exact}$ has such an expansion. $\endgroup$
    – origimbo
    Feb 21, 2018 at 13:24
  • $\begingroup$ I mean globally, for the scalar functional, not elementwise. The finite element space is Lagrange P1, the most basic one... $\endgroup$ Feb 21, 2018 at 19:28
  • $\begingroup$ In your first equation, I think you mean $h$ on the right hand side. $\endgroup$ Feb 22, 2018 at 23:39
  • $\begingroup$ @WolfgangBangerth: Yes, thank you for pointing that out. $\endgroup$ Feb 23, 2018 at 12:04

1 Answer 1

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The question you ask is simpler than you may think. For any mesh size $h$, you can compute $\lambda(h)$. If this dependence is continuous, then you can do a Taylor expansion of $\lambda(h)$ around $h=0$ to obtain exactly the form you want.

The only question in your case is whether $\lambda(h)$ is a continuous function. This is made slightly more complicated than one may think because in fact not all values of $h$ are of course possible, but at least for sufficiently small $h$, you can evaluate $\lambda$ for essentially any $h$. In that case, let us assume that $h$ can be chosen arbitrarily, and then the question can be reduced to the following: If $\lambda$ is a function of the solution $u(h)$, it is continuous if $u(h)$ is continuous as a function of $h$. For most problems, $u(h)$ is indeed continuous, and in that case $\lambda(h)$ is a continuous function if $\lambda(u)$ is continuous. Whether that is true is typically a functional analysis question: if $\lambda(u)$ is, for example, the value of $u$ at a particular point, then for $H^1$ functions $u$, the function $\lambda(u)$ is not continuous. But if $\lambda(u)$ is the average value of $u$, then the dependence is continuous.

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  • $\begingroup$ Thank you for your answer. In my case the function $\lambda(h)$ is the first eigenvalue of an elliptic operator (Laplace-Beltrami). It verifies something like $-\Delta_h u_h = \lambda_h u_h$, where $u_h$ is non-negative (the first eigenfunction) and $L^2$ normalized. I guess in this case $\lambda(h)$ should be regular enough for the development to work... $\endgroup$ Feb 23, 2018 at 12:10
  • $\begingroup$ @BeniBogosel -- yes, at least if the eigenvalue is single (i.e., has multiplicity one) I would expect that to be true. $\endgroup$ Feb 24, 2018 at 20:43
  • $\begingroup$ Do you have any reference in mind for your statement: "if this dependence is continuous, then you can do a Taylor expansion around 0"? Thank you. $\endgroup$ Mar 3, 2018 at 13:06
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    $\begingroup$ @feynman I see no reason why this argument couldn't be applied to finite differences just as well. But note that I made no assumption that a scheme actually converges: $\lambda(h=0)$ is simply the limit as the mesh is made finer and finer. I'm not assuming that $\lambda(0)=\lambda_\text{exact}$. $\endgroup$ Aug 6, 2023 at 12:18
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    $\begingroup$ @feynman I don't understand the question. No "reasonable" FD scheme diverges; no reasonable FE scheme diverges either. If a scheme diverges, then just don't use it -- it's not good for anything. The same is true for a scheme that converges, but to something that is not the solution. $\endgroup$ Aug 7, 2023 at 10:29

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