#### Document Type

Article

#### Publication Date

5-24-2011

#### Publication Title

SIAM Journal on Numerical Analysis

#### Department

Department of Mathematics

#### Abstract

We study eigenfunctions $\phi_j$ and eigenvalues $E_j$ of the Dirichlet Laplacian on a bounded domain $\Omega\subset\mathbb{R}^n$ with piecewise smooth boundary. We bound the distance between an arbitrary parameter $E>0$ and the spectrum $\{E_j\}$ in terms of the boundary $L^2$-norm of a normalized trial solution *u* of the Helmholtz equation $(\Delta+E)u=0$. We also bound the $L^2$-norm of the error of this trial solution from an eigenfunction. Both of these results are sharp up to constants, hold for all *E* greater than a small constant, and improve upon the best-known bounds of Moler–Payne by a factor of the wavenumber $\sqrt{E}$. One application is to the solution of eigenvalue problems at high frequency, via, for example, the method of particular solutions. In the case of planar, strictly star-shaped domains we give an inclusion bound where the constant is also sharp. We give explicit constants in the theorems, and show a numerical example where an eigenvalue around the 2500th is computed to 14 digits of relative accuracy. The proof makes use of a new quasi-orthogonality property of the boundary normal derivatives of the eigenmodes (Theorem 1.3), of interest in its own right. Namely, the operator norm of the sum of rank 1 operators $\partial_n\phi_j\langle\partial_n\phi_j,\cdot\rangle$ over all $E_j$ in a spectral window of width $\sqrt{E}$—a sum with about $E^{(n-1)/2}$ terms—is at most a constant factor (independent of *E*) larger than the operator norm of any one individual term.

#### DOI

10.1137/100796637

#### Dartmouth Digital Commons Citation

Barnett, A. H. and Hassell, A., "Boundary Quasi-Orthogonality and Sharp Inclusion Bounds for Large Dirichlet Eigenvalues" (2011). *Dartmouth Scholarship*. 2046.

https://digitalcommons.dartmouth.edu/facoa/2046