A Posteriori Error Analysis for Evolution Nonlinear Schrödinger Equations up to the Critical Exponent

Abstract

We provide a posteriori error estimates in the L([0, T]; L(?))-norm for relaxation time discrete and fully discrete schemes for a class of evolution nonlinear Schrödinger equations up to the critical exponent. In particular for the discretization in time we use the relaxation Crank–Nicolson-type scheme introduced by Besse in [SIAM J. Numer. Anal., 42 (2004), pp. 934–952]. The space discretization consists of finite element spaces that are allowed to change between time steps. The estimates are obtained using the reconstruction technique. Through this technique the problem is converted to a perturbation of the original partial differential equation and this makes it possible to use nonlinear stability arguments as in the continuous problem. Our analysis includes as special cases the cubic and quintic nonlinear Schrödinger equations in one spatial dimension and the cubic nonlinear Schrödinger equation in two spatial dimensions. Numerical results illustrate that the estimates are indeed of optimal order of convergence.