On the Weyl's law for discretized elliptic operators

Abstract

In this paper we give an estimate on the asymptotic behavior of eigenvalues of discretized elliptic boundary values problems. We first prove a simple min-max principle for selfadjoint operators on a Hilbert space. Then we show two sided bounds on the $k$-th eigenvalue of the discrete Laplacian by the $k$-th eigenvalue of the continuous Laplacian operator under the assumption that the finite element mesh is quasi-uniform. Combining this result with the well-known Weyl's law, we show that the $k$-th eigenvalue of the discretized isotropic elliptic operators, spectrally equivalent to the discretized Laplacian, is $\mathcal O\left(k^{2/d}\right)$. Finally, we show how these results can be used to obtain an error estimate for finite element approximations of elliptic eigenvalue problems.