Two-grid methods for a class of nonlinear elliptic eigenvalue problems

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

1 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique
2 IFSTTAR/COSYS/LISIS - Laboratoire Instrumentation, Simulation et Informatique Scientifique
3 LJLL - Laboratoire Jacques-Louis Lions

In this article, we introduce and analyse some two-grid methods for nonlinear elliptic eigenvalue problems of the form div(Du)+Vu+f(u2)u=»u,%u%L2=1. We provide a priori error estimates for the ground state energy, the eigenvalue » and the eigenfunction u, in various Sobolev norms. We focus in particular on the Fourier spectral approximation (for periodic boundary conditions), and on the P1 and P2 finite element discretizations (for homogeneous Dirichlet boundary conditions), taking numerical integration errors into account. Finally, we provide numerical examples illustrating our analysis.

Mots clés: NONLINEAR EIGENVALUE PROBLEM; SPECTRAL AND PSEUDO SPECTRAL APPROXIMATION; FINITE ELEMENT APPROXIMATION; GROUND STATE COMPUTATION; TWO GRID METHOD

Mots clés: APPROXIMATION; ANALYSE NUMERIQUE

Domaines: Analyse numérique [cs.NA]

Eric Cancès, Rachida Chakir, Linhua He, Yvon Maday. Two-grid methods for a class of nonlinear elliptic eigenvalue problems. IMA Journal of Numerical Analysis, 2018, 38 (2), pp.605-645. ⟨10.1093/imanum/drw053⟩. ⟨hal-01652289⟩