Mathematical models involved in the determination of electronic structures often require to solve nonlinear eigenvalue problems. These problems are computationally very costly, and therefore approximations have to be resorted to, among which the chosen model, the chosen discretization, and the chosen (possibly iterative) algorithm. The a priori analysis for the discretization of such problems 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.

In this talk, we shall present an a posteriori analysis for a one-dimensional Gross-Pitaevskii type equation in a periodic setting using a planewave discretization. We will provide a computable bound of the error between the exact and approximate solutions. 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. We shall also discuss the possibility of extending this theory to Gross-Pitaevskii equations with a magnetic field.

References:
[1] E.Cancès, R. Chakir and Y. Maday, Numerical analysis of the planewave discretization of orbital-free and Kohn-Sham models, M2AN (highlight article) 46 (2012) 341-388.
[2] G. Dusson, and Y. Maday, A posteriori analysis of a non-linear Gross-Pitaevskii type Eigenvalue Problem, submitted.