Boundary quasi-orthogonality and sharp inclusion bounds for large Dirichlet eigenvalues
Provides improved theoretical bounds for high-frequency eigenvalue problems, benefiting numerical methods like the method of particular solutions.
The authors derive sharp bounds for the distance between a parameter E and the Dirichlet spectrum, and for the L2 error of trial solutions, improving previous bounds by a factor of √E. They demonstrate 14-digit relative accuracy for the 2500th eigenvalue in a numerical example.
We study eigenfunctions and eigenvalues of the Dirichlet Laplacian on a bounded domain $Ω\subset\RR^n$ 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 $(Δ+ E)u = 0$. 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, of interest in its own right.