A posteriori error control & adaptivity for Crank-Nicolson finite element approximations for the linear Schrödinger equation
This work provides rigorous error control and adaptivity for numerical simulations of Schrödinger equations, benefiting computational scientists studying quantum dynamics.
The authors derive optimal order a posteriori error estimates for Crank-Nicolson finite element approximations of linear Schrödinger equations, and develop an adaptive algorithm that reduces computational cost substantially while providing efficient error control, especially for small Planck constants.
We derive optimal order a posteriori error estimates for fully discrete approximations of linear Schrödinger-type equations, in the $L^\infty(L^2)-$norm. For the discretization in time we use the Crank-Nicolson method, while for the space discretization we use finite element spaces that are allowed to change in time. The derivation 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 implementations 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 computational cost substantially and provides efficient error control for the solution and the observables of the problem, especially for small values of the Planck constant.