Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Optimal $L^2$ error estimates of mass- and energy-conserved FE schemes for a nonlinear Schrödinger-type system

Zhuoyue Zhang, Wentao Cai

TL;DR

This work develops an implicit Crank–Nicolson finite element scheme for nonlinear Schrödinger–type systems, including Schrödinger–Helmholz and Schrödinger–Poisson models, on convex/polyhedral domains. The method preserves discrete mass and energy, with existence and uniqueness of fully discrete solutions shown via Schaefer's fixed point theorem. It provides optimal $L^2$ error estimates for both models by an error-splitting analysis and Ritz projection techniques, validated by numerical experiments. The results demonstrate stable, structure-preserving discretization with convergence rates matching the theoretical predictions, offering a practical approach for simulations of Schrödinger-type phenomena.

Abstract

In this paper, we present an implicit Crank-Nicolson finite element (FE) scheme for solving a nonlinear Schrödinger-type system, which includes Schrödinger-Helmholz system and Schrödinger-Poisson system. In our numerical scheme, we employ an implicit Crank-Nicolson method for time discretization and a conforming FE method for spatial discretization. The proposed method is proved to be well-posedness and ensures mass and energy conservation at the discrete level. Furthermore, we prove optimal $L^2$ error estimates for the fully discrete solutions. Finally, some numerical examples are provided to verify the convergence rate and conservation properties.

Optimal $L^2$ error estimates of mass- and energy-conserved FE schemes for a nonlinear Schrödinger-type system

TL;DR

This work develops an implicit Crank–Nicolson finite element scheme for nonlinear Schrödinger–type systems, including Schrödinger–Helmholz and Schrödinger–Poisson models, on convex/polyhedral domains. The method preserves discrete mass and energy, with existence and uniqueness of fully discrete solutions shown via Schaefer's fixed point theorem. It provides optimal error estimates for both models by an error-splitting analysis and Ritz projection techniques, validated by numerical experiments. The results demonstrate stable, structure-preserving discretization with convergence rates matching the theoretical predictions, offering a practical approach for simulations of Schrödinger-type phenomena.

Abstract

In this paper, we present an implicit Crank-Nicolson finite element (FE) scheme for solving a nonlinear Schrödinger-type system, which includes Schrödinger-Helmholz system and Schrödinger-Poisson system. In our numerical scheme, we employ an implicit Crank-Nicolson method for time discretization and a conforming FE method for spatial discretization. The proposed method is proved to be well-posedness and ensures mass and energy conservation at the discrete level. Furthermore, we prove optimal error estimates for the fully discrete solutions. Finally, some numerical examples are provided to verify the convergence rate and conservation properties.
Paper Structure (10 sections, 6 theorems, 102 equations, 1 figure, 4 tables)

This paper contains 10 sections, 6 theorems, 102 equations, 1 figure, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries and the main results
  3. Discrete conservation laws
  4. Existence and uniqueness of numerical solution
  5. Optimal $L^2$ error estimates
  6. Error estimates of the Schrödinger--Helmholz equations
  7. Error estimates for the Schrödinger--Poisson equations
  8. Numerical Examples
  9. Conclusions
  10. Acknowledgments

Key Result

Theorem 2.1

Suppose that the system i1-i3 has a unique solution $(u,\phi)$ satisfying the regularity condition solution. Then there exists a positive constant $\tau_0$, such that when $\tau\leq\tau_0$, the fully-discrete system CN1-CN2 admits a unique FE solution $(u_h^n, \phi_h^n)$ satisfying where $C_0^*$ is a positive constant independent of $n$, $h$ and $\tau$.

Figures (1)

  • Figure 1: Evolution of discrete energy $\mathcal{E}_h^n$ and mass $\mathcal{M}_h^n$

Theorems & Definitions (13)

  • Theorem 2.1
  • Lemma 2.1
  • Theorem 3.1
  • proof
  • Remark 3.1
  • Lemma 4.1
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • proof
  • ...and 3 more