# Vérification des problèmes de valeurs propres

13

Commençons par un problème de forme

$\left(\mathcal{L}+{k}^{2}\right)u=0$

avec un ensemble de conditions aux limites données ( Dirichlet , Neumann , Robin , Periodic , Bloch-Periodic ). Cela correspond à la recherche des valeurs propres et des vecteurs propres pour un opérateur , sous une géométrie et des conditions aux limites. On peut obtenir un problème comme celui-ci en acoustique, électromagnétisme, élastodynamique, mécanique quantique, par exemple.$\mathcal{L}$$\mathcal{L}$

Je sais que l'on peut discrétiser l'opérateur en utilisant différentes méthodes, par exemple, les méthodes de différences finies pour obtenir

$\left[A\right]\left\{U\right\}={k}^{2}\left\{U\right\}$

or using, Finite Element Methods to obtain

$\left[K\right]\left\{U\right\}={k}^{2}\left[M\right]\left\{U\right\}\phantom{\rule{.5em}{0ex}}.$

In one case getting an eigenvalue problem and a generalized eigenvalue problem in the other. After obtaining the discrete version of the problem one uses a solver for the eigenvalue problem.

## Some thoughts

• The method of Manufactured Solutions is not useful in this case since there is no source term to balance the equation.
• One can verify that the matrices $\left[K\right]$$[K]$ and $\left[M\right]$$[M]$ are well captured using a frequency domain problem with source term, e.g.

$\left[{\mathrm{\nabla }}^{2}+{\omega }^{2}/{c}^{2}\right]u\left(\omega \right)=f\left(\omega \right)\phantom{\rule{.5em}{0ex}},\phantom{\rule{1em}{0ex}}\mathrm{\forall }\omega \in \left[{\omega }_{min},{\omega }_{max}\right]$

$\left[{\mathrm{\nabla }}^{2}+{k}^{2}\right]u=0\phantom{\rule{.5em}{0ex}}.$

But this will not check the solver issues.

• Maybe, one can compare solutions for different methods, like FEM and FDM.

## Question

What is the way to verify the solutions (eigenvalue-eigenvector pairs) for discretization schemes due to numerical methods like FEM and FDM for eigenvalue problems?

Can you compare your results to the spectra for known cases (square, cube, circle, sphere)? There are also expected convergence rates for eigenvectors and eigenvalues in appropriate norms that you can check (though these rates tend to vary depending on frequency - see journals.cambridge.org/action/…)
Jesse Chan

Yes, you can compare with analytic solutions. But normally they are provided for really simple cases. The question is about how to do the verification process. If there is something similar to the method oh manufactured solutions. Or if you should combine this method for other problems with analytical solutions.
nicoguaro

1
In one dimension, if you start with desired $k,v$$k,v$, and have $\left(\mathcal{L}+{k}^{2}\right)v=w\ne 0$$(\mathcal{L}+k^2)v = w \neq0$, you could try to decompose $w=fv+g{v}^{\prime }$$w = fv+gv'$, if such $f,g$$f,g$ exist, and then run with ${\mathcal{L}}^{\prime }=\mathcal{L}-f-g\mathrm{\partial }$$\mathcal{L}'=\mathcal{L}-f-g\partial$. This can mess up $\mathcal{L}$$\mathcal{L}$'s symmetries and other properties, I suppose. Here $v$$v$ and ${v}^{\prime }$$v'$ should be linearly independent, and can't vanish at the same point.
Kirill

@JesseChan, thanks for the suggested reading. It took me some time but I read it. I don't think that they provide enough information for the desired purpose.
nicoguaro

1
Je veux être sûr d'avoir bien compris. Voulez-vous savoir comment estimer la distance entre les paires propres calculées pour l'opérateur discret (matrice ou matrices) et la paire propre correspondante pour l'opérateur lisse? Ou voulez-vous maintenant comment estimer la précision avec laquelle vous avez résolu un problème de valeurs propres discrètes?
Carl Christian

Réponses:

3

I realize this question is old, but I just saw it and find it interesting. In the past, I have followed the suggestions found in this question's comments, coupled with some slightly more complicated cases that I'm familiar with in the literature (Orr--Sommerfeld is always handy).

However, I'm also aware of some literature on the inhomogeneous eigenvalue problems that arise when constructing a manufactured solution. There is some discussion of such problems here: DOI: 10.1016. These authors also suggest a so-called Method of Manufactured Cross Sections (MXS, I guess) to avoid this issue altogether, which I won't pretend to understand at the moment, but could very well be useful.

What they propose as "inhomogeneous eigenvalue problem" is the approach I proposed in my original post. I am still trying to understand the Method of Manufactured Cross Sections, though.
nicoguaro

I realize that, just suggesting that some literature exists for such problems so it might not be a dead-end as you suggested: "Manufactured Solutions is not useful in this case since there is no source term to balance the equation."
Spencer Bryngelson

It is not a criticism of your post. Quite the opposite! I am just commenting what I found after reading the reference to promote the discussion.
nicoguaro

2

For the second-order derivative (and the Laplacian on simple domains), expressions for the discrete eigenpairs (i.e. after discretization) are available. For example, for finite-difference, the eigenpairs are listed here.

Expression for the eigenpairs with a finite-element discretization can be found similarly (for P1 and P2 discretization).

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