A note on optimal $$H^1$$ H 1 -error estimates for Crank-Nicolson approximations to the n
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A note on optimal H 1 -error estimates for Crank-Nicolson approximations to the nonlinear Schrödinger equation Patrick Henning1
· Johan Wärnegård1
Received: 7 July 2019 / Accepted: 27 May 2020 © The Author(s) 2020
Abstract 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. Keywords Nonlinear Schrödinger equations · Finite elements · A priori error estimates · Energy conserving methods Mathematics Subject Classification 35Q55 · 65M60 · 65M15 · 81Q05
1 Introduction In this paper we consider nonlinear Schrödinger equations (NLS) seeking a complex function u(t, x) such that
Communicated by Mechthild Thalhammer. The authors acknowledge the support by the Swedish Research Council (Grant 2016-03339) and the Göran Gustafsson foundation.
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Patrick Henning [email protected] Johan Wärnegård [email protected]
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Department of Mathematics, KTH Royal Institute of Technology, 100 44 Stockholm, Sweden
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P. Henning, J. Wärnegård
i∂t u = −u + V u + γ (|u|2 )u in a bounded domain D ⊂ Rd , with a homogenous Dirichlet boundary condition on ∂D and a given initial value. Here, V (x) is a known real-valued potential and γ : [0, ∞) → R is a smooth (and possibly nonlinear) function that depends on the unknown density |u|2 . Of particular interest are cubic nonlinearities of the form γ (|u|2 )u = κ|u|2 u, for some κ ∈ R. In this case, the equation is called Gross– Pitaevskii equation. It has applications in optics [1,2], fluid dynamics [3,4] and, most importantly, in quantum physics, where it models for example the dynamics of BoseEinstein condensates in a magnetic trapping potential [5–7]. Another relevant class are saturated nonlinearities, such as γ (|u|2 ) = κ|u|2 (1 + α|u|2 )−1 for some α ≥ 0, which appear in the context of nonlinear optical wave propagation in layered metallic structures [8,9] or the propagation of light beams in plasmas [10]. In order to discretize nonlinear Schrödinger equations in time, splitting methods and exponential integrators yield typically highly efficient solution schemes that can be easily combined with a spectral discretization in space (cf. [11–18] and the references therein). If the exact solution to the NLS admits high regularity, such discretization schemes typically show a remarkably good performance. However, if the regularity of the solution is strongly reduced, either by rough potentials V (e.g. disorder potentials or optical l
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