High accuracy analysis of the lowest order H1-Galerkin mixed finite element method for nonlinear sine-Gordon equations

Shi, Dongyang; Wang, Fenling; Zhao, Yanxing

doi:10.1007/s10255-017-0692-z

Cited by 7 publications

(4 citation statements)

References 7 publications

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“…Assume I h be the corresponding interpolation operator on V h0 . It can be found in Reference [19] that if u ∈ H 3 (Ω), there holds…”

Section: Superconvergence Resultsmentioning

confidence: 99%

Superconvergence analysis of an energy stable scheme with three step backward differential formula‐finite element method for nonlinear reaction–diffusion equation

Wang

2021

Numerical Methods Partial

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A three step backward differential formula scheme is proposed for nonlinear reaction-diffusion equation and superconvergence results are studied with Galerkin finite element method unconditionally. Energy stability is testified for the constructed scheme with an artificial term. Splitting technique is utilized to get rid of the ratio between the time step size 𝜏 and the subdivision parameter h. Temporal error estimate in H 2 -norm is derived, which leads to the boundedness of the solutions of the time-discrete equations. Unconditional spatial error estimate in L 2 -norm is deduced which help bound the numerical solutions in L ∞ -norm. Superconvergent property of u n in H 1 -norm with order O(h 2 + 𝜏 3 ) is obtained by taking difference between two time levels of the error equations unconditionally. The global superconvergent property is deduced through the above results. Two numerical examples show the validity of the theoretical analysis.

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“…Assume I h be the corresponding interpolation operator on V h0 . It can be found in Reference [19] that if u ∈ H 3 (Ω), there holds…”

Section: Superconvergence Resultsmentioning

confidence: 99%

Superconvergence analysis of an energy stable scheme with three step backward differential formula‐finite element method for nonlinear reaction–diffusion equation

Wang

2021

Numerical Methods Partial

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“…Then, the classical Ritz projection operator R h is introduced and the unconditional error estimate R h U n − U n h 0 with the order O(h 2 ) is obtained, which imply that U n h is unconditionally bounded in the sense of L ∞ −norm. Furthermore, we arrive at the superclose property of R h U n −U n h 1 with the order O(τ 2 + h 2 ), combining with which and the relation between R h and the corresponding interpolation operator I h [13], the error I h u n −U n h 1 with the order O(τ 2 +h 2 ) holds unconditionally. Besides, the global superconvergence result is obtained by virtue of the interpolated postprocessing technique.…”

mentioning

confidence: 75%

Unconditional superconvergence analysis of a linearized Crank–Nicolson Galerkin FEM for generalized Ginzburg–Landau equation

Shi

Wang

2020

Computers & Mathematics with Applications

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In this paper, a linearized Crank-Nicolson Galerkin finite element method (FEM) for generalized Ginzburg-Landau equation (GLE) is considered, in which, the difference method in time and the standard Galerkin FEM are employed. Based on the linearized Crank-Nicolson difference method in time and the standard Galerkin finite element method with bilinear element in space, the time-discrete and space-time discrete systems are both constructed. We focus on a rigorous analysis and consideration of unconditional superconvergence error estimates of the discrete schemes. Firstly, by virtue of the temporal error results, the regularity for the time-discrete system is presented. Secondly, the classical Ritz projection is used to obtain the spatial error with order O(h 2 ) in the sense of L 2 −norm. Thanks to the relationship between the Ritz projection and the interpolated projection, the superclose estimate with order O(τ 2 + h 2 ) in the sense of H 1 −norm is derived. Thirdly, it follows from the interpolated postprocessing technique that the global superconvergence result is deduced. Finally, some numerical results are provided to confirm the theoretical analysis.Keywords Nonlinear Ginzburg-Landau equation · Finite element method · Linearized Crank-Nicolson scheme · Ritz projection and interpolated operators · Unconditional superconvergence results IntroductionIn this paper, we are concerned with the numerical solution of the following generalized GLE

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“…Let I h : H 2 ( ) → V 0h be the associated interpolation operator satisfying I h u(a i ) = u(a i ), where a i , (i = 1, 2, 3, 4) are the four vertices of K m . Imitating the proof given for [35,Lemma 2] yields…”

Section: Superconvergence Of the L1 Femmentioning

confidence: 87%

Superconvergence of a finite element method for the time-fractional diffusion equation with a time-space dependent diffusivity

2020

Adv Differ Equ

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In this work, a time-fractional diffusion problem with a time-space dependent diffusivity is considered. The solution of such a problem has a weak singularity at the initial time $t=0$ t = 0 . Based on the L1 scheme in time on a graded mesh and the conforming finite element method in space on a uniform mesh, the fully discrete L1 conforming finite element method (L1 FEM) of a time-fractional diffusion problem is investigated. The error analysis is based on a nonstandard discrete Gronwall inequality. The final superconvergence result shows that an optimal grading of the temporal mesh should be selected as $r\geq (2-\alpha )/\alpha $ r ≥ ( 2 − α ) / α . Numerical results confirm that our analysis is sharp.

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High accuracy analysis of the lowest order H1-Galerkin mixed finite element method for nonlinear sine-Gordon equations

Cited by 7 publications

References 7 publications

Superconvergence analysis of an energy stable scheme with three step backward differential formula‐finite element method for nonlinear reaction–diffusion equation

Superconvergence analysis of an energy stable scheme with three step backward differential formula‐finite element method for nonlinear reaction–diffusion equation

Unconditional superconvergence analysis of a linearized Crank–Nicolson Galerkin FEM for generalized Ginzburg–Landau equation

Superconvergence of a finite element method for the time-fractional diffusion equation with a time-space dependent diffusivity

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