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We provide a posteriori error estimates in the $L^\infty(L^2)-$norm forrelaxation time discrete and fully discrete schemes for a class of evolutionnonlinear Schr\"odinger equations up to the critical exponent. In particularfor the discretization in time we use the relaxation Crank-Nicolson-type schemeintroduced by Besse in \cite{Besse}. For the discretization in space we usefinite element spaces that are allowed to change between time steps. Theestimates are obtained using the reconstruction technique. Through thistechnique the problem is converted to a perturbation of the original partialdifferential equation and this makes it possible to use nonlinear stabilityarguments as in the continuous problem. In particular, main ingredients we usein our analysis are the Gagliardo-Nirenberg inequality and the two conservationlaws (mass and energy conservation) of the continuous problem. Numericalresults illustrate that the estimates are indeed of optimal order ofconvergence.

A posteriori error analysis for evolution nonlinear Schr\"odinger equations up to the critical exponent