A posteriori error control and adaptivity for Crank-Nicolson finite element approximations for the linear Schrödinger equation

We derive optimal order a posteriori error estimates for fully discrete approximations of linear Schrödinger-type equations, in the L∞(L2)-norm. For the discretization in time we use the Crank–Nicolson method, while for the space dis- cretization we use finite element spaces that are allowed to change in time. The deriva- tion of the estimators is based on a novel elliptic reconstruction that leads to estimates which reflect the physical properties of Schrödinger equations. The final estimates are obtained using energy techniques and residual-type estimators. Various numerical experiments for the one-dimensional linear Schrödinger equation in the semiclassical regime, verify and complement our theoretical results. The numerical implementa- tions are performed with both uniform partitions and adaptivity in time and space. For adaptivity, we further develop and analyze an existing time-space adaptive algorithm to the cases of Schrödinger equations. The adaptive algorithm reduces the computa