On the Weyl's law for discretized elliptic operators

TL;DR

This paper analyzes the asymptotic behavior of eigenvalues for discretized elliptic boundary value problems, establishing bounds and convergence rates that connect discrete spectra with continuous Weyl's law.

Contribution

It provides new bounds on eigenvalues of discretized elliptic operators and links these to Weyl's law, improving understanding of finite element approximations.

Findings

Eigenvalues of discretized elliptic operators grow as O(k^{2/d})

Two-sided bounds relate discrete and continuous eigenvalues

Error estimates for finite element eigenvalue approximations

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.