A note on optimal H1-error estimates for Crank-Nicolson approximations to the nonlinear Schrödinger equation

• Published in: BIT Vol. 61; no. 1; pp. 37 - 59
• Main Authors: Henning, Patrick, Wärnegård, Johan
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.03.2021; Springer Nature B.V
• Subjects: A priori error estimates; Computational Mathematics and Numerical Analysis; Energy conserving methods; Finite elements; Hilbert space; Iterative methods; Mathematical analysis; Mathematics; Mathematics and Statistics; Nonlinear Schrödinger equations; Nonlinear systems; Numeric Computing; Schrodinger equation; 65M15; Finite elements; 81Q05; A priori error estimates; 35Q55; Energy conserving methods; 65M60; Nonlinear Schrödinger equations
• ISSN: 0006-3835; 1572-9125; 1572-9125
• DOI: 10.1007/s10543-020-00814-3

In this paper we consider a mass- and energy–conserving Crank-Nicolson time discretization for a general class of nonlinear Schrödinger equations. This scheme, which enjoys popularity in the physics community due to its conservation properties, was already subject to several analytical and numerical studies. However, a proof of optimal L ∞ ( H 1 ) -error estimates is still open, both in the semi-discrete Hilbert space setting, as well as in fully-discrete finite element settings. This paper aims at closing this gap in the literature. We also suggest a fixed point iteration to solve the arising nonlinear system of equations that makes the method easy to implement and efficient. This is illustrated by numerical experiments.