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(u2)u=λu, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\|u\|_{L^{2}}=1$\end{document}. 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\|u_{\delta}-u\|_{H^{1}}$\end{document}, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\|u_{\delta}-u\|_{L^{2}}$\end{document}, |λδ−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, |λδ−λ| goes to zero as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\|u_{\delta}-u\|_{H^{1}}^{2}+\|u_{\delta}-u\|_{L^{2}}$\end{document}. We also show that under more restrictive assumptions on A, V and f, |λδ−λ| converges to zero as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\|u_{\delta}-u\|_{H^{1}}^{2}$\end{document}, 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.