Applied/PDE Seminar: Virginie Ehrlacher (CERMICS - ENPC, Paris), 'Periodic Schrödinger operators with local defects and spectral pollution '

Event Date: 

Friday, November 16, 2012 -
3:00pm to 4:00pm

Event Location: 

  • 4607B South Hall

Event Contact: 

Carlos Garcia-Cervera
Email: cgarcia@math.ucsb.edu

(Host: C. Garcia-Cervera)

In this talk, we will deal with the numerical calculation of eigenvalues of perturbed periodic Schrödinger operators located in spectral gaps.

Such operators are encountered in the modeling of the electronic structure of crystals with local defects, and of photonic crystals. The usual finite element Galerkin approximation is known to give rise to spectral pollution. We will give a precise description of the corresponding spurious states, which can be interpreted as surface states. We then prove that the supercell model does not produce spectral pollution and provide some a priori error estimates. In particular, in the absence of numerical integration, we prove that the rate of convergence is exponential with respect to the size of the supercell. Lastly, we extend results by Lewin and Séré on some no-pollution criteria. In particular, we prove that using approximate spectral projectors enables one to eliminate spectral pollution in a given spectral gap of the reference periodic Schödinger operator. In practice, this method may be viewed as an Augmented Finite Element Method where special basis functions, computed with the so-called Wannier functions, must be added to the standard Finite Element basis. We will show numerical simulations which will illustrate these three results.