Finite element eigenvalue enclosures for the Maxwell operator

We propose employing the extension of the Lehmann-Maehly-Goerisch method developed by Zimmermann and Mertins, as a highly effective tool for the pollution-free finite element computation of the eigenfrequencies of the resonant cavity problem on a bounded region. This method gives complementary bounds for the eigenfrequencies which are adjacent to a given real parameter. We present a concrete numerical scheme which provides certified enclosures in a suitable asymptotic regime. We illustrate the applicability of this scheme by means of some numerical experiments on benchmark data using Lagrange elements and unstructured meshes.Comment: arXiv admin note: substantial text overlap with arXiv:1306.535

Local two-sided bounds for eigenvalues of self-adjoint operators

We examine the equivalence between an extension of the Lehmann-Maehly-Goerisch method developed a few years ago by Zimmermann and Mertins, and a geometrically motivated method developed more recently by Davies and Plum. We establish a general framework which allows sharpening various previously known results in these two settings and determine explicit convergence estimates for both methods. We demonstrate the applicability of the method of Zimmermann and Mertins by means of numerical tests on the resonant cavity problem

The foundations of spectral computations via the Solvability Complexity Index hierarchy: Part I

The problem of computing spectra of operators is arguably one of the most investigated areas of computational mathematics. Recent progress and the current paper reveal that, unlike the finite-dimensional case, infinite-dimensional problems yield a highly intricate infinite classification theory determining which spectral problems can be solved and with which type of algorithms. Classifying spectral problems and providing optimal algorithms is uncharted territory in the foundations of computational mathematics. This paper is the first of a two-part series establishing the foundations of computational spectral theory through the Solvability Complexity Index (SCI) hierarchy and has three purposes. First, we establish answers to many longstanding open questions on the existence of algorithms. We show that for large classes of partial differential operators on unbounded domains, spectra can be computed with error control from point sampling operator coefficients. Further results include computing spectra of operators on graphs with error control, the spectral gap problem, spectral classifications, and discrete spectra, multiplicities and eigenspaces. Second, these classifications determine which types of problems can be used in computer-assisted proofs. The theory for this is virtually non-existent, and we provide some of the first results in this infinite classification theory. Third, our proofs are constructive, yielding a library of new algorithms and techniques that handle problems that before were out of reach. We show several examples on contemporary problems in the physical sciences. Our approach is closely related to Smale's program on the foundations of computational mathematics initiated in the 1980s, as many spectral problems can only be computed via several limits, a phenomenon shared with the foundations of polynomial root finding with rational maps, as proved by McMullen

A HIERARCHICAL METHOD FOR OBTAINING EIGENVALUE ENCLOSURES

Abstract. We introduce a new method of obtaining guaranteed enclosures of the eigenvalues of a variety of self-adjoint differential and difference operators with discrete spectrum. The method is based upon subdividing the region into a number of simpler regions for which eigenvalue enclosures are already available. 1