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

TL;DR

This paper develops a posteriori error estimates for numerical schemes solving evolution nonlinear Schrödinger equations up to the critical exponent, using reconstruction techniques and stability arguments.

Contribution

It introduces a novel a posteriori error analysis framework for fully discrete schemes of nonlinear Schrödinger equations up to the critical exponent, incorporating adaptive finite element discretizations.

Findings

Error estimates are of optimal order of convergence.

The analysis successfully handles changing finite element spaces.

Numerical results confirm the theoretical error bounds.

Abstract

We provide a posteriori error estimates in the $L^\infty(L^2)-$norm for relaxation time discrete and fully discrete schemes for a class of evolution nonlinear Schr\"odinger 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 \cite{Besse}. For the discretization in space we use 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. In particular, main ingredients we use in our analysis are the Gagliardo-Nirenberg inequality and the two conservation laws (mass and energy conservation) of the continuous problem. Numerical results illustrate that the estimates are indeed of optimal order of convergence.