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I have to approximate the smallest eigenvalue of the following generalized eigenvalue problem

$$ - \nabla \cdot D(x) \nabla p(x) + \alpha(x) p(x) = \lambda \beta(x) p(x) $$

over a domain like follows (many more subdomains, this is only an approximation to illustrate the problem)

Domain

The eigenvalues are all positive, so to approximate the smallest one I invert the left side of the equation after the spatial discretization, and use Arnoldi method. The functions $D(x)$, $\alpha(x)$ and $\beta(x)$ are piecewise constant over the whole domain, being probably different over different subdomains.

I already have a code to approximate the smallest eigenvalue and associated eigenvector with a Continuous Galerkin FEM.

I was wondering which locally conservative method can I use to solve the eigenvalue problem (i.e., I want the solution to satisfy the balance equation over each subdomain).

What I know:


OPTION 1: I know that using discontinuous Galerkin FEM the balance equation will be satisfied over each subdomain, but it usually deal with the mixed formulation of the problem

$$ - \nabla \cdot q(x) + \alpha(x) p(x) = \lambda \beta(x) p(x) \\ q(x) - D(x) \nabla p(x) = 0 $$

and it introduces a lot of degrees of freedom to the system.


OPTION 2: I have read that it can also be done using a Hybridizable discontinuous Galerkin, because the we can apply a static condensation technique and the degrees of freedom related to $q$ are the only ones that we have to solve in a coupled way. It works fine for source problems, but I think that we can not apply this technique with an eigenvalue problem.

edit: (why static condensation will not work for eigenproblems?) If dealing with a source problem, we form the linear system for the hybridizable discontinuous Galerkin as follows:

$$ \left[\begin{array}{c c} A & B \\ C & D \end{array}\right] \left[\begin{array}{c} p \\ q \end{array}\right] = \left[\begin{array}{c} r \\ s \end{array}\right] $$

We know that matrix $A$ is block diagonal because the degrees of freedom are completelly decoupled (from element to element), so we perform the static condensation to solve the following problem

$$ (D-CA^{-1}B)q = s-CA^{-1} r $$

and then we recover $p$ with

$$ Ap = r-Bq $$

When we form the algebraic eigenvalue problem with discontinuous galerking we get

$$ \left[\begin{array}{c c} A & B \\ C & D \end{array}\right] \left[\begin{array}{c} p \\ q \end{array}\right] = \lambda \left[\begin{array}{c c} E & 0 \\ 0 & 0 \end{array}\right] \left[\begin{array}{c} p \\ q \end{array}\right] $$

Then, as in a linear system, we try to isolate $p$ in the first equation, and we obtain

$$ p = (A-\lambda E)^{-1}B q $$

and after substitution in the second equation we get

$$ (D + C(A-\lambda E)^{-1}B ) q = 0 $$

Matrices $A$ and $E$ are block diagonal, so easy to invert, but the eigenvalue $\lambda$ now is inside the matrix to invert, so the eigenvalue problem has been converted into a nonlinear eigenvalue problem. Different linear approximations of $(A-\lambda E)^{-1}$ (taylor expansion on $\lambda = 0$ or $\lambda = \lambda_0$, for instance) will solve the problem of the nonlinear eigenvalue problem, but then the algorithm will converge much more slower and succesive corrections will be necessaries. Thus static condensation is not a good idea here.


OPTION 3: There are some papers that talk about locally conservative continuos Galerkin methods: The Continuous Galerkin Method Is Locally Conservative, and Locally Conservative Fluxes for the Continuous Galerkin Method. The first one seems to perform some extra calculations to approximate the fluxes, that for my is equivalent to the mixed formulation. And the second one says that is posible to deal with the primal formulation, but also develop the article about a formulation that includes the fluxes $q(x)$.


QUESTION: it is no clear for me that locally conservation can be obtained with a primal formultaion (like in the first equation I posted), moreover taking into accout that we are solving an eigenvalue problem. Can it be done?

Any idea or advice about literature on the topic would be greatly appreciated. I am mainly interested in FEM methods, but any idea is welcome.

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    $\begingroup$ There isn't really any magic with eigenvalue problems. What makes you think that certain methods can not be used for eigenvalues problems? $\endgroup$ – Wolfgang Bangerth Feb 24 '14 at 5:31
  • $\begingroup$ Hi @WolfgangBangerth. I have updated the post to explain why Hybridizable Discontinuous Galerkin can not be used. I know that usual Discontinuous Galerkin will work, but I was wondering if there is any known (not known by me, but known by others) continuous Galerkin approach that is locally conservative, or any way to avoid the extra degrees of feedom from discontinuous galerkin. Thanks. $\endgroup$ – sebas Feb 24 '14 at 16:39
  • $\begingroup$ In the hybridizable method, you would solve the generalized eigenvalue problem in the outer loop, and as part of the method you need to invert the matrix on the left which you can do as you would always do. Just don't try to eliminate one of the variables before solving the eigenvalue problem. Eliminate it in every iteration. $\endgroup$ – Wolfgang Bangerth Feb 25 '14 at 16:33
  • $\begingroup$ Hi @WolfgangBangerth, this is a good idea. I am trying to go a bit farther. The point is that when solving a linear system, if static condensation is applied, it actually has an effect on the theretical size of the problem (from dimension $n$ to dimension $m$, with $n>m$). I mean, gmres will recover the exact solution after $m$ iterations (it will also have an effect into the number of iterations to converge to a desired tolerance). $\endgroup$ – sebas Feb 25 '14 at 16:54
  • $\begingroup$ If I can reduce the dimension of the eigenproblem, then it will need less Arnoldi iterations to converge, if I only use the static condensation to solve the systems, it will be the same number of systems to solve (even if they are solved faster than before). $\endgroup$ – sebas Feb 25 '14 at 16:54

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