A posteriori error analysis for evolution nonlinear Schrödinger equations up to the critical exponent
For researchers in numerical analysis of PDEs, this work offers rigorous error control for a class of nonlinear Schrödinger equations, though it is an incremental extension of existing techniques.
The paper provides a posteriori error estimates for relaxation time discrete and fully discrete schemes for nonlinear Schrödinger equations up to the critical exponent, using the reconstruction technique and nonlinear stability arguments. Numerical results confirm optimal order of convergence.
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 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 \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.