Numerical analysis of nonlinear eigenvalue problems

Abstract

We provide a priori error estimates for variational approximations of the ground state eigenvalue and eigenvector of nonlinear elliptic eigenvalue problems of the form $-div(A\nabla u)+Vu+f(u^2)u=\lambda u$, $\Vert u{\Vert }_{L^2}=1$. 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 $(u_{\delta },{\lambda }_{\delta })$ a variational approximation of the ground state eigenpair $(u,\lambda )$, we are interested in the convergence rates of $\Vert u_{\delta }-u{\Vert }_{H^1}$, $\Vert u_{\delta }-u{\Vert }_{L^2}$ and $|{\lambda }_{\delta }-\lambda |$, when the discretization parameter $\delta$ goes to zero. We prove that if $A$, $V$ and $f$ satisfy certain conditions, $|{\lambda }_{\delta }-\lambda |$ goes to zero as $\Vert u_{\delta }-u{\Vert }_{H^1}^2+\Vert u_{\delta }-u{\Vert }_{L^2}$. We also show that under more restrictive assumptions on $A$, $V$ and $f$, $|{\lambda }_{\delta }-\lambda |$ converges to zero as $\Vert u_{\delta }-u{\Vert }_{H^1}^2$, thus recovering a standard result for {\em linear} elliptic eigenvalue problems. For the latter analysis, we make use of estimates of the error $u_{\delta }-u$ in negative Sobolev norms.

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