Startseite Optimal Convergence of the Newton Iterative Crank–Nicolson Finite Element Method for the Nonlinear Schrödinger Equation
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Optimal Convergence of the Newton Iterative Crank–Nicolson Finite Element Method for the Nonlinear Schrödinger Equation

  • Hanzhang Hu , Buyang Li und Jun Zou EMAIL logo
Veröffentlicht/Copyright: 25. Mai 2022
Computational Methods in Applied Mathematics
Aus der Zeitschrift Computational Methods in Applied Mathematics Band 22 Heft 3

Abstract

An error estimate is presented for the Newton iterative Crank–Nicolson finite element method for the nonlinear Schrödinger equation, fully discretized by quadrature, without restriction on the grid ratio between temporal step size and spatial mesh size. It is shown that the Newton iterative solution converges double exponentially with respect to the number of iterations to the solution of the implicit Crank–Nicolson method uniformly for all time levels, with optimal convergence in both space and time.

Keywords: Nonlinear Schrödinger Equation; Crank–Nicolson; Newton Iteration; Quadrature; Convergence
MSC 2010: 65N12; 65N15; 35Q55

Funding source: Natural Science Foundation of Guangdong Province

Award Identifier / Grant number: 2018A0303100016

Funding source: Research Grants Council, University Grants Committee

Award Identifier / Grant number: 15301818

Award Identifier / Grant number: 14306921

Award Identifier / Grant number: 14306719

Funding source: Hong Kong Polytechnic University

Award Identifier / Grant number: P0031035

Funding statement: The work of the first author was supported by Natural Science Foundation of Guangdong province, China (2018A0303100016). The work of the second author was partially supported by Hong Kong RGC General Research Fund (project 15301818) and an internal grant of the university (Project ID: P0031035, Work Programme: ZZKQ). The work of the third author was substantially supported by Hong Kong RGC General Research Fund (projects 14306921 and 14306719).

Appendix: Proof of Lemma 4.1

(i) By using the definition in (4.1) and Hölder’s inequality, we have

∥ v h ∥ H h s = ∥ ( - Δ h ) s 2 ⁢ v h ∥ L 2 = ( ∑ j = 1 J | ( v h , ϕ j ) | 2 - s ⁢ λ j s ⁢ | ( v h , ϕ j ) | s ) 1 2 ≤ ( ∑ j = 1 J | ( v h , ϕ j ) | 2 ) 2 - s 4 ⁢ ( ∑ j = 1 J λ j 2 ⁢ | ( v h , ϕ j ) | 2 ) s 4 = ∥ v h ∥ L 2 2 - s 2 ⁢ ∥ Δ h ⁢ v h ∥ L 2 s 2 = ∥ v h ∥ L 2 1 - s 2 ⁢ ∥ v h ∥ H h 2 s 2 .

(ii) By the definition of the discrete Laplacian operator, we have

| ( Δ h ⁢ u h , v ) | = | ( Δ h ⁢ u h , P h ⁢ v ) | = | ( ∇ ⁡ u h , ∇ ⁡ P h ⁢ v ) | ≤ C ⁢ ∥ ∇ ⁡ u h ∥ L 2 ⁢ ∥ ∇ ⁡ P h ⁢ v ∥ L 2 ≤ C ⁢ h - 2 ⁢ ∥ u h ∥ L 2 ⁢ ∥ P h ⁢ v ∥ L 2 ≤ C ⁢ h - 2 ⁢ ∥ u h ∥ L 2 ⁢ ∥ v ∥ L 2   for all ⁢ v ∈ L 2 ⁢ ( ( Ω ) ) ,

where the second to last inequality is the standard inverse inequality for finite element functions. Since the inequality above holds for all v ∈ L 2 ⁢ ( ( Ω ) ) , it follows that ∥ Δ h ⁢ u h ∥ L 2 ≤ C ⁢ h - 2 ⁢ ∥ u h ∥ L 2 . By using this estimate and the interpolation inequality proved in Lemma 4.1 (i), we have

∥ ( - Δ h ) s 2 ⁢ v h ∥ L 2 = ∥ v h ∥ H h s ≤ ∥ v h ∥ L 2 2 - s 2 ⁢ ∥ v h ∥ H h 2 s 2 = ∥ v h ∥ L 2 2 - s 2 ⁢ ∥ Δ h ⁢ v h ∥ L 2 s 2 ≤ C ⁢ h - s ⁢ ∥ v h ∥ L 2 .

Since ( - Δ h ) s 2 2 ⁢ v h = ( - Δ h ) s 2 - s 1 2 ⁢ ( - Δ h ) s 1 2 ⁢ v h , the inequality above implies that

∥ ( - Δ h ) s 2 2 ⁢ v h ∥ L 2 ≤ C ⁢ h - ( s 2 - s 1 ) ⁢ ∥ ( - Δ h ) s 1 2 ⁢ v h ∥ L 2 .

This proves the second result of Lemma 4.1.∎

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Received: 2022-03-10
Accepted: 2022-03-10
Published Online: 2022-05-25
Published in Print: 2022-07-01

© 2022 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

  1. Frontmatter
  2. On the Spectrum of an Operator Associated with Least-Squares Finite Elements for Linear Elasticity
  3. Anisotropic Adaptive Finite Elements for an Elliptic Problem with Strongly Varying Diffusion Coefficient
  4. DPG Methods for a Fourth-Order div Problem
  5. Stable Implementation of Adaptive IGABEM in 2D in MATLAB
  6. Optimal Convergence of the Newton Iterative Crank–Nicolson Finite Element Method for the Nonlinear Schrödinger Equation
  7. Partially Discontinuous Nodal Finite Elements for 𝐻(curl) and 𝐻(div)
  8. Improvement of the Constructive A Priori Error Estimates for a Fully Discretized Periodic Solution of Heat Equation
  9. An 𝐿𝑝-DPG Method with Application to 2D Convection-Diffusion Problems
  10. An Improvement on a Class of Fixed Point Iterative Methods for Solving Absolute Value Equations
  11. Low-Regularity Integrator for the Davey–Stewartson System: Elliptic-Elliptic Case
  12. The Numerical Approximation to a Stochastic Age-Structured HIV/AIDS Model with Nonlinear Incidence Rates
  13. Qualitative Properties of Space-Dependent SIR Models with Constant Delay and Their Numerical Solutions
  14. A Staggered Discontinuous Galerkin Method for Quasi-Linear Second Order Elliptic Problems of Nonmonotone Type
https://doi.org/10.1515/cmam-2022-0057
Schlagwörter für diesen Artikel
Nonlinear Schrödinger Equation; Crank–Nicolson; Newton Iteration; Quadrature; Convergence

Artikel in diesem Heft

  1. Frontmatter
  2. On the Spectrum of an Operator Associated with Least-Squares Finite Elements for Linear Elasticity
  3. Anisotropic Adaptive Finite Elements for an Elliptic Problem with Strongly Varying Diffusion Coefficient
  4. DPG Methods for a Fourth-Order div Problem
  5. Stable Implementation of Adaptive IGABEM in 2D in MATLAB
  6. Optimal Convergence of the Newton Iterative Crank–Nicolson Finite Element Method for the Nonlinear Schrödinger Equation
  7. Partially Discontinuous Nodal Finite Elements for 𝐻(curl) and 𝐻(div)
  8. Improvement of the Constructive A Priori Error Estimates for a Fully Discretized Periodic Solution of Heat Equation
  9. An 𝐿𝑝-DPG Method with Application to 2D Convection-Diffusion Problems
  10. An Improvement on a Class of Fixed Point Iterative Methods for Solving Absolute Value Equations
  11. Low-Regularity Integrator for the Davey–Stewartson System: Elliptic-Elliptic Case
  12. The Numerical Approximation to a Stochastic Age-Structured HIV/AIDS Model with Nonlinear Incidence Rates
  13. Qualitative Properties of Space-Dependent SIR Models with Constant Delay and Their Numerical Solutions
  14. A Staggered Discontinuous Galerkin Method for Quasi-Linear Second Order Elliptic Problems of Nonmonotone Type
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