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I am trying to find the first few eigenvalues of an operator defined by the following PDE:

$$ \begin{cases} -\Delta u +(1-\varphi)u=\lambda u, & \text{ on }\Omega = [0,1]^2 \\ u=0 & \text{ on } \partial \Omega \end{cases}.$$

I have used a finite element method to discretize the problem, and I am now faced with a (large) matrix eigenvalue problem. In the equation above $\varphi$ is in fact a characteristic function.

If $0<\lambda_1(\Omega)\leq \lambda_2(\Omega)\leq ...$ is the sequence of eigenvalues of my operator, then what eigenvalues of the discretized matrix should I look at to recover $\lambda_1(\Omega),\lambda_2(\Omega)$, etc, and their corresponding eigenvectors?

[edit] As I've seen from some numerical tests most likely the smallest eigenvalue of the matrix is the closest to $\lambda_1(\Omega)$, the second smallest to $\lambda_2(\Omega)$ and so on.

Is there some result which says that as the dimension of the matrix increases, the corresponding matrix eigenvalues converge to the actual eigenvalues of the operator?

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From : This course on spectral mesh processing page 31 of the course, their advice is to combine the Arnoldi iterations provided by ARPACK with a shifting method to compute efficiently the eigenvalues band by band. This is about how to compute the eigenvalues numerically, but the question about the convergence remains.

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