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

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+ u(3) = lambda u, parallel to u parallel to(2)(L) = 1, with periodic boundary conditions in one dimension. Denoting by (u(N), lambda(N)) the variational approximation of the ground state eigenpair (u, lambda) based on a Fourier spectral approximation and by (u(N)(k), lambda(k)(N)) 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 parallel to u -u(N)parallel to(1)(H), parallel to u -u(N)parallel to(2)(L), vertical bar lambda-lambda(N)vertical bar and then present original a posteriori estimates of the convergence rates of parallel to u -u(N)(k)parallel to(1)(H) when N and k go to infinity. We introduce a residual representing the global error R-N(k) = -Delta u(N)(k) + Vu(N)(k) +(u(N)(k))(3) -lambda(k)(N)u(N)(k) 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.