Numerical Analysis of Nonlinear Eigenvalue Problems

We provide a priori error estimates for variational approximations of the ground state eigenvalue and eigenvector of nonlinear elliptic eigenvalue problems of the form −div(A∇u) + V u + f(u)u = λu, ‖u‖L2 = 1. We focus in particular on the Fourier spectral approximation (for periodic problems) and on the P1 and P2 finite-element discretizations. Denoting by (uδ, λδ) a variational approximation of the ground state eigenpair (u, λ), we are interested in the convergence rates of ‖uδ − u‖H1 , ‖uδ − u‖L2 and |λδ − λ|, when the discretization parameter δ goes to zero. We prove that if A, V and f satisfy certain conditions, |λδ − λ| goes to zero as ‖uδ − u‖ 2 H1 + ‖uδ − u‖L2 . We also show that under more restrictive assumptions on A, V and f , |λδ − λ| converges to zero as ‖uδ − u‖ 2 H1 , thus recovering a standard result for linear elliptic eigenvalue problems. For the latter analysis, we make use of estimates of the error uδ − u in negative Sobolev norms.