Numerical analysis of nonlinear eigenvalue problems

Eric Cancès (1, 2) , Rachida Chakir (3) , Yvon Maday (3)

1 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique
2 MICMAC - Methods and engineering of multiscale computing from atom to continuum
3 LJLL - Laboratoire Jacques-Louis Lions

We provide a priori error estimates for variational approximations of the ground state eigenvalue and eigenvector of nonlinear elliptic eigenvalue problems . We focus in particular on the Fourier spectral approximation (for periodic problems) and on the P₁ and P₂ finite-element discretizations. Our analysis extends to the case of nonlinear eigenproblems the classical results about the comparative speeds of convergence of the eigenvalues with respect to the eigenvectors in the H¹- norm. We show that under some assumptions we recover a standard result for linear elliptic eigenvalue problems.

Domaines: Analyse numérique [cs.NA]

Eric Cancès, Rachida Chakir, Yvon Maday. Numerical analysis of nonlinear eigenvalue problems. Journal of Scientific Computing, 2010, 45 (1-3), pp.90-117. ⟨10.1007/s10915-010-9358-1⟩. ⟨hal-00392025⟩