Numerical Analysis of Nonlinear Eigenvalue Problems

Abstract

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 ℙ₁ and ℙ₂ 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 $\|u_{\delta}-u\|_{H^{1}}$, $\|u_{\delta}-u\|_{L^{2}}$, |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 $\|u_{\delta}-u\|_{H^{1}}^{2}+\|u_{\delta}-u\|_{L^{2}}$. We also show that under more restrictive assumptions on A, V and f, |λ _δ −λ| converges to zero as $\|u_{\delta}-u\|_{H^{1}}^{2}$, 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.