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Linearly Implicit Finite Element Methods Approximating the Solution to the Nonlinear Schrödinger Equation with a
Panagiotis Paraschis1,2, 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.
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.
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