Verificação em problemas de autovalor

Vamos começar com um problema do formulário

(L + k^{2}) u = 0

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

com um conjunto de determinadas condições de contorno ( Dirichlet , Neumann , Robin , Periodic , Bloch-Periodic ). Isso corresponde à localização dos valores próprios e dos vetores próprios para algum operador , sob alguma geometria e condições de contorno. Pode-se obter um problema como esse em acústica, eletromagnetismo, elastodinâmica, mecânica quântica, por exemplo. $\mathcal{L}$

Eu sei que é possível discretizar o operador usando métodos diferentes, por exemplo, Métodos de Diferenças Finitas para obter

[A] {U} = k^{2} {U}

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

ou usando métodos de elementos finitos para obter

[K] {U} = k^{2} [M] {U} .

$[K]\{U\} = k^2 [M]\{U\} \enspace .$

Em um caso, obter um problema de autovalor e um problema generalizado de autovalor no outro. Após obter a versão discreta do problema, é utilizado um solucionador para o problema de autovalor.

Alguns pensamentos

O método das Soluções Manufaturadas não é útil nesse caso, pois não há termo de origem para equilibrar a equação.
Pode-se verificar se as matrizes $[K]$ e $[M]$ são bem capturadas usando um problema no domínio da frequência com o termo fonte, por exemplo

$[\nabla^{2} + ω^{2} / c^{2}] u (ω) = f (ω), \forall ω \in [ω_{min}, ω_{max}]$ $[\nabla^2 + \omega^2/c^2] u(\omega) = f(\omega) \enspace ,\quad \forall \omega \in [\omega_\min, \omega_\max]$
instead of

$[\nabla^{2} + k^{2}] u = 0 .$ $[\nabla^2 + k^2] u = 0 \enspace .$
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?

— nicoguaro
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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

In one dimension, if you start with desired

k, v

$k,v$ , and have

(L + k^{2}) v = w \neq 0

$(\mathcal{L}+k^2)v = w \neq0$ , you could try to decompose

w = f v + g v^{'}

$w = fv+gv'$ , if such

f, g

$f,g$ exist, and then run with

L^{'} = L - f - g \partial

$\mathcal{L}'=\mathcal{L}-f-g\partial$ . This can mess up

L

$\mathcal{L}$ 's symmetries and other properties, I suppose. Here

v

$v$ and

v^{'}

$v'$ 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

I want to be sure that I have understood you correctly. Do you want to know how to estimate the distance between computed eigenpairs for discrete operator (matrix or matrices) and the corresponding eigenpair for the smooth operator? Or do you want to now how to estimate the accuracy by which you have solved a discrete eigenvalue problem?

— Carl Christian

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.

— Spencer Bryngelson
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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 # 4/19

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

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).

— user7440
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