Abstract. I calculate the modes of vibration of membranes of arbitrary shape using a collocation approach based on Little Sinc Functions. The matrix representation of the PDE obtained using this method is explicit and it does not require the calculation of integrals. To illustrate the virtues of this approach, I have considered a large number of examples, part of them taken from the literature, and part of them new. When possible, I have tested the accuracy of these results by comparing them with the exact results (when available) or with results from the literature. In particular, in the case of the L-shaped membrane, the first example discussed in the paper, I show that it is possible to extrapolate the results obtained with different grid sizes to obtain higly precise results. Finally, I also show that the present collocation technique can be easily combined with conformal mapping to provide numerical approximations to the energies which quite rapidly converge to the exact results.

Abstract. The Dirichlet eigenvalue or “drum ” problem in a domain Ω ⊂ R 2 becomes numerically challenging at high eigenvalue (frequency) E. In this regime the method of particular solutions (MPS) gives spectral accuracy for many domain shapes. It requires a number of degrees of freedom scaling as √ E, the number of wavelengths on the boundary, in contrast to direct discretization for which this scaling is E. Our main result is an inclusion bound on eigenvalues that is a factor O ( √ E)tighter than the classical bound of Moler–Payne and that is optimal in that it reflects the true slopes of curves appearing in the MPS. We also present an MPS variant that cures a normalization problem in the original method, while evaluating basis functions only on the boundary. This method is efficient at high frequencies, where we show that, in practice, our inclusion bound can give three extra digits of eigenvalue accuracy with no extra effort.

Abstract. We study eigenfunctions φj and eigenvalues Ej of the Dirichlet Laplacian on a bounded domain Ω ⊂ 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 ( ∆ + 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 √ 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 below), of interest in its own right. Namely, the operator norm of the sum of rank 1 operators ∂nφj〈∂nφj, · 〉 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.

We consider a large class of residuum based a posteriori eigenvalue/eigenvector estimates and present an abstract framework for proving their asymptotic exactness. Equivalence of the estimator and the error is also established. To demonstrate the strength of our abstract approach we present a detailed study of hierarchical error estimators for Laplace eigenvalue problems in planar polygonal regions. To this end we develop new error analysis for the Galerkin approximation which avoids the use of the strengthened Cauchy-Schwarz inequality and the saturation assumption, and gives reasonable and explicitly computable upper bounds on the discretization error. A brief discussion is also given concerning the design of estimators which are in the same spirit, but are based on different a posteriori techniques—notably, those of gradient recovery type.