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

Hanzhang Hu; Buyang Li; Jun Zou

doi:10.1515/cmam-2022-0057

Article

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

Hanzhang Hu , Buyang Li and Jun Zou

Published/Copyright: May 25, 2022

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From the journal Computational Methods in Applied Mathematics Volume 22 Issue 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

\begin{matrix} {∥ v_{h} ∥}_{H_{h}^{s}}^{s} & = {∥ {(- Δ_{h})}^{\frac{s}{2}} v_{h} ∥}_{L^{2}}^{2} = {(J \sum j = 1 {| (v_{h}, ϕ_{j}) |}^{2 - s} λ_{j}^{s} {| (v_{h}, ϕ_{j}) |}^{s})}^{\frac{1}{2}} \leq {(J \sum j = 1 {| (v_{h}, ϕ_{j}) |}^{2})}^{\frac{2 - s}{4}} {(J \sum j = 1 λ_{j}^{2} {| (v_{h}, ϕ_{j}) |}^{2})}^{\frac{s}{4}} = {∥ v_{h} ∥}_{h}^{\frac{2 - s}{2}} {∥ Δ_{h} v_{h} ∥}_{L^{2}}^{\frac{s}{2}} = {∥ v_{h} ∥}_{h}^{1 - \frac{s}{2}} {∥ v_{h} ∥}_{h}^{\frac{s}{2}} . \end{matrix}

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

\begin{matrix} | (Δ_{h} u_{h}, v) | & = | (Δ_{h} u_{h}, P_{h} v) | = | (\nabla u_{h}, \nabla P_{h} v) | \leq C {∥ \nabla u_{h} ∥}_{L^{2}}^{2} {∥ \nabla P_{h} v ∥}_{L^{2}}^{2} \leq C h^{- 2} {∥ u_{h} ∥}_{L^{2}}^{2} {∥ P_{h} v ∥}_{L^{2}}^{2} \leq C h^{- 2} {∥ u_{h} ∥}_{L^{2}}^{2} {∥ v ∥}_{L^{2}}^{2} for all v \in L^{2} ((Ω)), \end{matrix}

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

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

{∥ {(- Δ_{h})}^{\frac{s}{2}} v_{h} ∥}_{L^{2}}^{2} = {∥ v_{h} ∥}_{H_{h}^{s}}^{s} \leq {∥ v_{h} ∥}_{h}^{\frac{2 - s}{2}} {∥ v_{h} ∥}_{h}^{\frac{s}{2}} = {∥ v_{h} ∥}_{h}^{\frac{2 - s}{2}} {∥ Δ_{h} v_{h} ∥}_{L^{2}}^{\frac{s}{2}} \leq C h^{- s} {∥ v_{h} ∥}_{L^{2}}^{2} .

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

Since ${(- Δ_{h})}^{\frac{s_{2}}{2}} v_{h} = {(- Δ_{h})}^{\frac{s_{2} - s_{1}}{2}} {(- Δ_{h})}^{\frac{s_{1}}{2}} v_{h}$ $( - Δ h ) s 2 2 ⁢ v h = ( - Δ h ) s 2 - s 1 2 ⁢ ( - Δ h ) s 1 2 ⁢ v h$ , the inequality above implies that

{∥ {(- Δ_{h})}^{\frac{s_{2}}{2}} v_{h} ∥}_{L^{2}}^{2} \leq C h^{- (s_{2} - s_{1})} {∥ {(- Δ_{h})}^{\frac{s_{1}}{2}} v_{h} ∥}_{L^{2}}^{2} .

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

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

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DOI: doi.org/10.1515/cmam-2022-0057

Keywords for this article

Nonlinear Schrödinger Equation; Crank–Nicolson; Newton Iteration; Quadrature; Convergence