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

Theodoros Katsaounis, Irene Kyza (Lead / Corresponding author)

    Research output: Contribution to journalArticle

    6 Citations (Scopus)

    Abstract

    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-
    Original languageEnglish
    Pages (from-to)55-90
    Number of pages36
    JournalNumerische Mathematik
    Volume129
    Issue number1
    Early online date9 May 2014
    DOIs
    Publication statusPublished - Jan 2015

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