## Abstract

We study eigenfunctions fj and eigenvalues φj of the Dirichlet Laplacian on a bounded domain Ω C ℝ^{n} with piecewise smooth boundary. We bound the distance between an arbitrary parameter E >0 and the spectrum {Ej} 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 vE. 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 δnφj [δnφj over all Ej in a spectral window of width √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.

Original language | English |
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Pages (from-to) | 1046-1063 |

Number of pages | 18 |

Journal | SIAM Journal on Numerical Analysis |

Volume | 49 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2011 |