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Related Experiment Video

Updated: May 29, 2026

Lens-free Video Microscopy for the Dynamic and Quantitative Analysis of Adherent Cell Culture

Lens-free Video Microscopy for the Dynamic and Quantitative Analysis of Adherent Cell Culture

Published on: February 23, 2018

Linearly Implicit Finite Element Methods Approximating the Solution to the Nonlinear Schrödinger Equation with a

Panagiotis Paraschis^1,2, Georgios E Zouraris³

¹Faculty of Mathematics, University of Vienna, Oscar-Morgestern-Platz 1, A-1090 Vienna, Austria.

Journal of Scientific Computing

|May 28, 2026

Summary

This study analyzes numerical methods for a nonlinear Schrödinger equation, presenting optimal error estimates for the Linearized Backward Euler finite element (LBEFE) and Linearized Crank-Nicolson finite element (LCNFE) methods. Numerical experiments validate the performance of these dissipative and conservative approaches.

Keywords:

Convergence Error estimates Finite element method Linearly implicit time-stepping Nonlinear Schrödinger equation Schamel nonlinearity Stability

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Area of Science:

Computational Mathematics
Numerical Analysis
Partial Differential Equations

Background:

Nonlinear Schrödinger equations are crucial in modeling various physical phenomena.
Accurate numerical solutions are essential for understanding complex behaviors.
Existing methods may have limitations in terms of accuracy, stability, or applicability across different dimensions.

Purpose of the Study:

To develop and analyze finite element methods for a nonlinear Schrödinger equation with specific boundary conditions.
To derive optimal error estimates in both L^2 and H^1 norms for the proposed numerical schemes.
To investigate the impact of dimensionality and time-step constraints on the accuracy of the methods.

Main Methods:

Application of the Linearized Backward Euler finite element (LBEFE) method, a dissipative scheme.
Application of the Linearized Crank-Nicolson finite element (LCNFE) method, a conservative scheme.
Derivation of error estimates using mathematical analysis, considering time step (τ) and spatial mesh width (h).

Main Results:

Optimal order error estimates of O(τ + h^2) in the L^2 norm for both LBEFE and LCNFE methods.
Error estimates of O(τ^α + h) in the H^1 norm, with α = 3/4 for LBEFE and α = 1/2 for LCNFE.
Identification of dimensionality-dependent mesh conditions for d=2, 3, while d=1 imposes no CFL conditions.

Conclusions:

Both LBEFE and LCNFE methods provide accurate approximations for the nonlinear Schrödinger equation.
The derived error estimates are optimal and depend on the chosen numerical scheme and spatial dimension.
Numerical experiments confirm the theoretical findings and demonstrate the practical performance of the methods.