On variational eigenvalue approximation of semidefinite operators

Eigenvalue problems for semidefinite operators with infinite dimensional kernels appear for instance in electromagnetics. Variational discretizations with edge elements have long been analyzed in terms of a discrete compactness property. As an alternative, we show here how the abstract theory can be developed in terms of a geometric property called the vanishing gap condition. This condition is shown to be equivalent to eigenvalue convergence and intermediate between two different discrete variants of Friedrichs estimates. Next we turn to a more practical means of checking these properties. We introduce a notion of compatible operator and show how the previous conditions are equivalent to the existence of such operators with various convergence properties. In particular the vanishing gap condition is shown to be equivalent to the existence of compatible operators satisfying an Aubin-Nitsche estimate. Finally we give examples demonstrating that the implications not shown to be equivalences, indeed are not.Comment: 26 page

Discrete compactness for the p-version of discrete differential forms

In this paper we prove the discrete compactness property for a wide class of p-version finite element approximations of non-elliptic variational eigenvalue problems in two and three space dimensions. In a very general framework, we find sufficient conditions for the p-version of a generalized discrete compactness property, which is formulated in the setting of discrete differential forms of any order on a d-dimensional polyhedral domain. One of the main tools for the analysis is a recently introduced smoothed Poincar\'e lifting operator [M. Costabel and A. McIntosh, On Bogovskii and regularized Poincar\'e integral operators for de Rham complexes on Lipschitz domains, Math. Z., (2010)]. For forms of order 1 our analysis shows that several widely used families of edge finite elements satisfy the discrete compactness property in p-version and hence provide convergent solutions to the Maxwell eigenvalue problem. In particular, N\'ed\'elec elements on triangles and tetrahedra (first and second kind) and on parallelograms and parallelepipeds (first kind) are covered by our theory

Discontinuous Galerkin approximation of the Maxwell eigenproblem

A theoretical framework for the analysis of discontinuous Galerkin approximations of the Maxwell eigenproblem with discontinuous coefficients is presented. Necessary and sufficient conditions for a spurious-free approximation are established, and it is shown that, at least on conformal meshes, basically all the discontinuous Galerkin methods in the literature actually fit into this framework. Relations with the classical theory for conforming approximations are also discussed

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

A mixed FEM for the quad-curl eigenvalue problem

The quad-curl problem arises in the study of the electromagnetic interior transmission problem and magnetohydrodynamics (MHD). In this paper, we study the quad-curl eigenvalue problem and propose a mixed method using edge elements for the computation of the eigenvalues. To the author's knowledge, it is the first numerical treatment for the quad-curl eigenvalue problem. Under suitable assumptions on the domain and mesh, we prove the optimal convergence. In addition, we show that the divergence-free condition can be bypassed. Numerical results are provided to show the viability of the method

On the Fattorini Criterion for Approximate Controllability and Stabilizability of Parabolic Systems

In this paper, we consider the well-known Fattorini's criterion for approximate controllability of infinite dimensional linear systems of type $y'=A y+Bu$. We precise the result proved by H. O. Fattorini in \cite{Fattorini1966} for bounded input $B$, in the case where $B$ can be unbounded or in the case of finite-dimensional controls. More precisely, we prove that if Fattorini's criterion is satisfied and if the set of geometric multiplicities of $A$ is bounded then approximate controllability can be achieved with finite dimensional controls. An important consequence of this result consists in using the Fattorini's criterion to obtain the feedback stabilizability of linear and nonlinear parabolic systems with feedback controls in a finite dimensional space. In particular, for systems described by partial differential equations, such a criterion reduces to a unique continuation theorem for a stationary system. We illustrate such a method by tackling some coupled Navier-Stokes type equations (MHD system and micropolar fluid system) and we sketch a systematic procedure relying on Fattorini's criterion for checking stabilizability of such nonlinear systems. In that case, the unique continuation theorems rely on local Carleman inequalities for stationary Stokes type systems