Dirichlet boundary conditions in generalized eigenvalue problem

Question

Let us consider a problem of the form

$$(\mathcal{L} + k^2) u(\mathbf{x})=0\, ,\quad \forall \mathbf{x} \in \Omega$$

with Dirichlet boundary conditions

$$u(\mathbf{x}) = 0, \quad \forall \mathbf{x} \in \partial\Omega\, .$$

We discretize this problem using, for example, the finite element method, to obtain the following generalized eigenvalue problem

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

To impose boundary conditions one can do one of the following:

1. Reduce the system by deleting rows/columns corresponding to boundary degrees of freedom in matrices $[K]$ and $[M]$.

2. Zero row/column values for the boundary degrees of freedom of both matrices $[K]$ and $[M]$, and fix the diagonal values in $[K]$ to any finite number. The problem with this approach is that the matrix $[M]$ loses its positive-definiteness (if it was) and eigensolvers do not like this.

3. Zero row/column values for the boundary degrees of freedom of both matrices $[K]$ and $[M]$, and fix the diagonal values in $[M]$ to any finite number.

For eigenvalue problems I have (almost) exclusively used 1, while I have used the other methods (or equivalent) for solving linear systems of equations.

Questions

• What is the most common method used for imposing Dirichlet boundary conditions?

• Is there any other method that I have not mentioned?

This question has been partially answered in here.

Answer 1

If $u$ is described spectrally, then there are other methods as I mentioned in that old answer. If not, I am unsure of additional options for you. This is a well-known problem that often results in spurious eigenvalues, with plenty of literature to back it up (e.g. see here). Most of it lies in the area of hydrodynamic stability, which might help guide your search, or at least assure yourself you are in good company.

From what I have seen, most people use approach 1. from your question if possible, though I prefer approach 2 as it is more general. If you're keen on maintaining positive-definiteness of $M$, then you might be able to modify option 2. to accommodate this: instead of zero-ing all the boundary values/rows you care about, you can constrain those rows to be such that $M$ remains PD, then modify the associated row of $K$ to be whatever it needs to be to satisfy your boundary condition. I have some concern that this might only be possible in the spectral space, since in that case all of the DOFs of $u$ would be associated with the BC (and so you have more options to play around with).