A nonconforming quadrilateral finite element approximation to nonlinear schrödinger equation

Shi, Dongyang; Liao, Xin; Wang, Lele

doi:10.1016/s0252-9602(17)30024-3

Cited by 10 publications

(4 citation statements)

References 18 publications

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“…With ( 29) and ( 37), the proof is completed. Finally, we use mathematical induction to verify assumption (15) which is similar to the technique used in [22,23,26].…”

Section: Superclose Analysis For the Fully-discrete Approximation Schemementioning

confidence: 99%

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High Accuracy Analysis of Nonconforming Mixed Finite Element Method for the Nonlinear Sivashinsky Equation

Wang

Liao

2020

Mathematical Problems in Engineering

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The fourth-order nonlinear Sivashinsky equation is often used to simulate a planar solid-liquid interface for a binary alloy. In this paper, we study the high accuracy analysis of the nonconforming mixed finite element method (MFEM for short) for this equation. Firstly, by use of the special property of the nonconforming EQ1rot element (see Lemma 1), the superclose estimates of order Oh2+Δt in the broken H1-norm for the original variable u and intermediate variable p are deduced for the back-Euler (B-E for short) fully-discrete scheme. Secondly, the global superconvergence results of order Oh2+Δt for the two variables are derived through interpolation postprocessing technique. Finally, a numerical example is provided to illustrate validity and efficiency of our theoretical analysis and method.

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“…With ( 29) and ( 37), the proof is completed. Finally, we use mathematical induction to verify assumption (15) which is similar to the technique used in [22,23,26].…”

Section: Superclose Analysis For the Fully-discrete Approximation Schemementioning

confidence: 99%

“…Meanwhile, superconvergence is critical in practical engineering numerical calculation and has always been a research hotspot. To find out more applications, readers may refer [12,[15][16][17][18][19][20][21][22][23]. As far as our knowledge is concerned, research on superconvergence for Sivashinsky equation is yet to be found.…”

Section: Introductionmentioning

confidence: 99%

High Accuracy Analysis of Nonconforming Mixed Finite Element Method for the Nonlinear Sivashinsky Equation

Wang

Liao

2020

Mathematical Problems in Engineering

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“…Along the analytical front, one can refer to [ 3 , 6 ] and the references therein. Along the numerical front, different efficient and accurate numerical methods including finite difference (FD) methods [ 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 ], finite element (FE) methods [ 18 , 19 , 20 , 21 , 22 ], spectral method [ 23 ], discontinuous Galerkin method [ 24 ], virtual element method [ 25 ], and so on. Furthermore, there have also been some other efficient algorithms, such as the multigrid methods [ 26 , 27 , 28 ] and two-grid methods [ 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 ], where the idea is presented firstly by Xu in [ 38 ].…”

Section: Introductionmentioning

confidence: 99%

A Time Two-Mesh Compact Difference Method for the One-Dimensional Nonlinear Schrödinger Equation

Liu

2022

Entropy

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The nonlinear Schrödinger equation is an important model equation in the study of quantum states of physical systems. To improve the computing efficiency, a fast algorithm based on the time two-mesh high-order compact difference scheme for solving the nonlinear Schrödinger equation is studied. The fourth-order compact difference scheme is used to approximate the spatial derivatives and the time two-mesh method is designed for efficiently solving the resulting nonlinear system. Comparing to the existing time two-mesh algorithm, the novelty of the new algorithm is that the fine mesh solution, which becomes available, is also used as the initial guess of the linear system, which can improve the calculation accuracy of fine mesh solutions. Compared to the two-grid finite element methods (or finite difference methods) for nonlinear Schrödinger equations, the numerical calculation of this method is relatively simple, and its two-mesh algorithm is implemented in the temporal direction. Taking advantage of the discrete energy, the result with O(τC4+τF2+h4) in the discrete L2-norm is obtained. Here, τC and τF are the temporal parameters on the coarse and fine mesh, respectively, and h is the space step size. Finally, some numerical experiments are conducted to demonstrate its efficiency and accuracy. The numerical results show that the new algorithm gives highly accurate results and preserves conservation laws of charge and energy. Furthermore, by comparing with the standard nonlinear implicit compact difference scheme, it can reduce the CPU time without loss of accuracy.

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“…[13] derived the same results as [12] for NLSE with the conforming linear triangular element by establishing the relationship between Ritz projection and the linear interpolation. Whereafter, a series of superconvergence results about backward Euler and Crank-Nicolson fully-discrete schemes for NLSE also were studied in [14][15][16][17].…”

Section: Introductionmentioning

confidence: 99%

Low order nonconforming finite element method for time-dependent nonlinear Schrödinger equation

Zhou

Shi

et al. 2018

Bound Value Probl

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The main aim of this paper is to apply a low order nonconforming EQ rot 1 finite element to solve the nonlinear Schrödinger equation. Firstly, the superclose property in the broken H 1-norm for a backward Euler fully-discrete scheme is studied, and the global superconvergence results are deduced with the help of the special characters of this element and the interpolation postprocessing technique. Secondly, in order to reduce computing cost, a two-grid method is developed and the corresponding superconvergence error estimates are obtained. Finally, a numerical experiment is carried out to confirm the theoretical analysis.

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A nonconforming quadrilateral finite element approximation to nonlinear schrödinger equation

Cited by 10 publications

References 18 publications

High Accuracy Analysis of Nonconforming Mixed Finite Element Method for the Nonlinear Sivashinsky Equation

High Accuracy Analysis of Nonconforming Mixed Finite Element Method for the Nonlinear Sivashinsky Equation

A Time Two-Mesh Compact Difference Method for the One-Dimensional Nonlinear Schrödinger Equation

Low order nonconforming finite element method for time-dependent nonlinear Schrödinger equation

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