# Verification in Eigenvalue problems

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

with a set of given boundary conditions (Dirichlet, Neumann, Robin, Periodic, Bloch-Periodic). This corresponds with finding the eigenvalues and eigenvectors for some operator $\mathcal{L}$, under some geometry, and boundary conditions. One can obtain a problem like this in acoustics, electromagnetism, elastodynamics, quantum mechanics, for example.

I know that one can discretize the operator using different methods, e.g, Finite Difference Methods to obtain

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

or using, Finite Element Methods to obtain

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

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 $[K]$ and $[M]$ are well captured using a frequency domain problem with source term, e.g.

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

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

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?

• 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 Jul 15 '15 at 4:03
• 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 Jul 15 '15 at 4:05
• In one dimension, if you start with desired $k,v$, and have $(\mathcal{L}+k^2)v = w \neq0$, you could try to decompose $w = fv+gv'$, if such $f,g$ exist, and then run with $\mathcal{L}'=\mathcal{L}-f-g\partial$. This can mess up $\mathcal{L}$'s symmetries and other properties, I suppose. Here $v$ and $v'$ should be linearly independent, and can't vanish at the same point. – Kirill Jul 15 '15 at 4:23
• @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 Aug 8 '15 at 0:45
• 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 Jun 8 '16 at 17:08