The numerical computation of the eigenvalues and eigenvectors of a self-adjoint operator on an infinite dimensional separable Hilbert space is a standard problem of numerical analysis and scientific computing, with a wide range of applications in science and engineering. Such problems are encountered in particular in mechanics (vibrations of elastic structures), electromagnetism and acoustics (resonant modes of cavities), and quantum mechanics (bound states of quantum systems).
Galerkin methods provide an efficient way to compute the discrete eigenvalues of a bounded-from-below self-adjoint operator A lying below the bottom of the essentialspectrum of A. On the other hand, Galerkin methods may fail to approximate discrete eigenvalues located in spectral gaps, that is between two points of the essentialspectrum of A. In some cases, the Galerkin method cannot find some of the eigenvalues of A located in spectral gaps (lack of approximation); in other cases, the limit set of the spectrum of the Galerkin approximations of A contains points which do not belong to the spectrum of A (spectral pollution). Such problems arise in various applications, such as the numerical simulation of photonic crystals, of doped semiconductors, or of heavy atoms with relativistic models.
Another aspect of this problem is the large nonlinearity and the complexity that arise in the models derived from the Schrödinger model. The numerical analysis of these nonlinear eigenvalue problems makes it possible to understand the convergence behavior, when it works but also when and why the numerical algorithm lacks optimality. From this analysis corrections can be proposed and implemented.
We will present recent results on the numerical analysis of these problems, obtained in collaboration with Rachida Chakir (University Paris 6) and Virginie Ehrlacher (Ecole des Ponts and INRIA).