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 del u) + Vu + f (u(2))u =lambda u, parallel to u parallel to(L2) = 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(delta), lambda(delta)) a variational approximation of the ground state eigenpair (u, lambda), we are interested in the convergence rates of parallel to u(delta) - u parallel to(H1), parallel to u(delta) - u parallel to(L2), |lambda(delta)-lambda.|, and the ground state energy, when the discretization parameter delta goes to zero. We prove in particular that if A, V and f satisfy certain conditions, |lambda(delta)-lambda vertical bar goes to zero as parallel to u(delta)-u parallel to(2)(H1) + parallel to u(delta)-u parallel to(L2). We also show that under more restrictive assumptions on A, V and f, |lambda(delta) - lambda| converges to zero as parallel to u(delta) - u parallel to(2)(H1), thus recovering a standard result for linear elliptic eigenvalue problems. For the latter analysis, we make use of estimates of the error u(delta-u) in negative Sobolev norms.