Numerical analysis of nonlinear eigenvalue problems

We provide a priori error estimates for variational approximations of the ground state energy, eigenvalue and eigenvector of nonlinear elliptic eigenvalue problems of the form -div(A ▽ u)+Vu+f(u ²)u=λ u, ∥u∥L ²=1 . We focus in particular on the Fourier spectral approximation (for periodic problems) and on the ℙ ₁ and ℙ ₂ finite-element discretizations. Denoting by (u _δ,λ _δ ) a variational approximation of the ground state eigenpair (u,λ), we are interested in the convergence rates of ∥uδ-u∥H ¹∥uδ-u∥L ², |λ _δ -λ|, and the ground state energy, when the discretization parameter δ goes to zero. We prove in particular that if A, V and f satisfy certain conditions, |λ _δ -λ| goes to zero as ∥uδ-u∥H ¹²+∥uδ- u∥ L ² . We also show that under more restrictive assumptions on A, V and f, |λ _δ -λ| converges to zero as ∥uδ-u∥H ¹², thus recovering a standard result for linear elliptic eigenvalue problems. For the latter analysis, we make use of estimates of the error u _δ -u in negative Sobolev norms.