A posteriori analysis of a nonlinear Gross-Pitaevskii-type eigenvalue problem

Abstract

In this article, we provide a first full a posteriori error analysis for variational approximations of the ground state eigenvector of a nonlinear elliptic problem of the Gross-Pitaevskii type, more precisely of the form $-Delta u + Vu + u3 = λu, |u|L3=1, with periodic boundary conditions in one dimension. Denoting by (uN,λN) the variational approximation of the ground state eigenpair $(u,λ) based on a Fourier spectral approximation and by $(uNk,λNk) the approximate solution at the kth iteration of an algorithm used to solve the nonlinear problem, we first provide a precise a priori analysis of the convergence rates of |u-uN|H1, |u-uN|L2, |λ-λN|and then present original a posteriori estimates of the convergence rates of |u-uNk|H1 when N and k go to infinity. We introduce a residual representing the global error RNk=-Delta uNk+VuNk+(uNk)3-λNkuNk and we divide it into two residuals characterizing, respectively, the error due to the discretization of space and the finite number of iterations when solving the problem numerically. Finally, in a series of numerical tests, we illustrate numerically the performance of this a posteriori analysis.