A posteriori error control & adaptivity for Crank-Nicolson finite element approximations for the linear Schr\"odinger equation

TL;DR

This paper develops optimal a posteriori error estimates and adaptive algorithms for Crank-Nicolson finite element approximations of the linear Schrödinger equation, improving efficiency and accuracy especially in the semiclassical regime.

Contribution

It introduces a novel elliptic reconstruction technique for error estimation and extends adaptive algorithms to Schrödinger equations, enhancing computational efficiency.

Findings

Error estimates accurately reflect physical properties of Schrödinger equations.

Adaptive algorithms significantly reduce computational cost.

Numerical experiments confirm theoretical error bounds and efficiency.

Abstract

We derive optimal order a posteriori error estimates for fully discrete approximations of linear Schr\"odinger-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 Schr\"odinger equations. The final estimates are obtained using energy techniques and residual-type estimators. Various numerical experiments for the one-dimensional linear Schr\"odinger 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 Schr\"odinger 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.