Joint work with Eric Cancès, Yvon Maday, Benjamin Stamm, and Martin Vohralík.

In this talk, I will present an a posteriori error analysis on the Gross-Pitaevskii equation. This problem requires to solve a nonlinear eigenvalue problem, which is computationally costly. Therefore approximations have to be resorted to, among which the chosen discretization, and the chosen (possibly iterative) algorithm.

The a priori analysis for the discretization of such a problem is quite recent (see e.g. [1]). It proves the convergence and the optimality of the method used. For a given approximation, the a posteriori analysis, once it is performed, provides a guaranteed upper bound on the total error. It may also enable to separate error components stemming from the different sources of approximation and control each of them. This makes possible to iteratively fit the approximation parameters leading to small errors at low computational cost.

After an introduction on the a posteriori error analysis, I will provide a computable upper bound of the error between the exact and approximate solutions of a one-dimensional Gross-Pitaevskii equation in a periodic setting using a planewave discretization. This bound will then be decomposed into two components, each of them depending mainly on one approximation parameter, which are here the number of degrees of freedom and the number of iterations in this iterative algorithm used to solve the problem numerically. Numerical simulations will be presented to illustrate the possibility of balancing the different error components.

(1) E. Cancès, R. Chakir, and Y. Maday, Numerical analysis of nonlinear eigenvalue problems, J. Sci. Comput. 45 (2010) 90–117.
(2) E. Cancès, G. Dusson, Y. Maday, B. Stamm, and M. Vohralík, A perturbation-method-based a posteriori estimator for the planewave discretization of nonlinear Schrödinger equations, Comptes Rendus Mathematique, 352(11), 941-946. (2014)