Unconditional Superconvergence Analysis for Nonlinear Parabolic Equation with $${\textit{EQ}}_1^{rot}$$ EQ 1 r o t Nonconforming Finite Element

Shi, Dongyang; Wang, Junjun; Yan, Fengna

doi:10.1007/s10915-016-0243-4

Cited by 72 publications

(35 citation statements)

References 30 publications

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“…Different from Shi et al, Gao, and Cai et al, 8,9,20,21 in this paper, we develop a linearized BDF-3 type scheme for nonlinear Sobolev equation and discuss unconditional superconvergence estimates with bilinear element. Because the specularity of the BDF-3 scheme, the particular inequality in Shi et al 8,9 cannot work, and some skills, such as transferring from one part in the inner product to the other, in Shi et al and Shi and Wang [11][12][13][14][15] [10][11][12][13][14][15] is exploited to get rid of the time restriction. The error u n − U n h is split into the temporal error u n − U n and the spatial error u n − U n h by a time-discrete system with solution U n .…”

Section: Introductionmentioning

confidence: 99%

Superconvergence analysis of a linearized three‐step backward differential formula finite element method for nonlinear Sobolev equation

Wang

2019

Math Methods in App Sciences

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A linearized three‐step backward differential formula (BDF) Galerkin finite element method (FEM) is developed for nonlinear Sobolev equation with bilinear element. Temporal error and spatial error are discussed through introducing a time‐discrete system. Solutions of the time‐discrete system are bounded in H2‐norm by the temporal error. Superconvergence results of order O(h2 + τ3) in H1‐norm for the original variable are deduced based on the spatial error. Some new tricks are utilized to get higher order of the temporal error and the spatial error. At last, two numerical examples are provided to support the theoretical analysis. Here, h is the subdivision parameter, and τ is the time step.

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Section: Introductionmentioning

confidence: 99%

Superconvergence analysis of a linearized three‐step backward differential formula finite element method for nonlinear Sobolev equation

Wang

2019

Math Methods in App Sciences

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“…Then the spatial error leads to the unconditional boundedness of a numerical solution in the L ∞ -norm. Subsequently, this so-called splitting technique was also applied to the other nonlinear parabolic type equations in [10][11][12][13][14][15][16][17][18]. Later, in [19] and [20] a second-order scheme for the nonlinear hyperbolic equation and the unconditional superconvergence analysis by using the splitting skill were given.…”

Section: Introductionmentioning

confidence: 99%

“…Such a practice can be used to avoid the difficulty in constructing a linearized scheme for a nonlinear hyperbolic equation and also give the error analysis for u and q = u t at the same time. Then we develop a linearized backward Euler FE scheme for the nonlinear parabolic system and apply the idea of splitting technique in [10][11][12][13][14][15][16][17][18][19][20] to split the error into the temporal and spatial errors. We obtain a temporal error, which implies the regularities of the solutions about the time-discrete equations.…”

Section: Introductionmentioning

confidence: 99%

A new approach to convergence analysis of linearized finite element method for nonlinear hyperbolic equation

Wang

Guo

2019

Bound Value Probl

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We study a new way to convergence results for a nonlinear hyperbolic equation with bilinear element. Such equation is transformed into a parabolic system by setting the original solution u as u t = q. A linearized backward Euler finite element method (FEM) is introduced, and the splitting skill is exploited to get rid of the restriction on the ratio between h and τ. The boundedness of the solutions about the time-discrete system in H 2-norm is proved skillfully through temporal error. The spatial error is derived without the mesh-ratio, where some new techniques are utilized to deal with the problems caused by the new parabolic system. The final unconditional optimal error results of u and q are deduced at the same time. Finally, a numerical example is provided to support the theoretical analysis. Here h is the subdivision parameter, and τ is the time step.

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“…A general way to treat this issue is employing the inverse inequality, which will result in some time-step restrictions, such as = O(h r + 1 )(r = 0, 1, 2) for Ginzburg-Landau equation in superconductivity [11,12]. Consequently, this technique has been widely applied to various nonlinear problems to obtain optimal error results or superconvergent properties [14][15][16][17][18][19][20]. Consequently, this technique has been widely applied to various nonlinear problems to obtain optimal error results or superconvergent properties [14][15][16][17][18][19][20].…”

mentioning

confidence: 99%

“…In order to overcome this difficulty, the splitting technique was used to solve a class of nonlinear parabolic equations [13]. Consequently, this technique has been widely applied to various nonlinear problems to obtain optimal error results or superconvergent properties [14][15][16][17][18][19][20]. However, all of the above works are focused on regular rectangle or triangle meshes.…”

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confidence: 99%

Nonconforming quadrilateral finite element method for Ginzburg–Landau equation

Shi

Liu

2019

Numerical Methods Partial

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Nonconforming finite element method is studied for a linearized backward fully-discrete scheme of the Ginzburg-Landau equation with the quadrilateral EQ rot 1 element. The unconditional convergent result of order O(h + ) in the broken H 1 -norm is deduced rigorously based on a splitting technique, by which the ratio between the subdivision parameter h and the time step is removed. Furthermore, numerical results are provided to confirm the theoretical analysis. The analysis developed herein can be regarded as a framework to deal with the unconditional convergent analysis of the Ginzburg-Landau equation for other known low order nonconforming elements. KEYWORDS EQ rot1 element, Galerkin method, Ginzburg-Landau equation, quadrilateral EQ rot 1 element, linearized scheme, quadrilateral meshes, unconditional convergent estimates INTRODUCTIONThe Ginzburg-Landau equation is an important nonlinear evolution equation, which can be used to describe a vast of phenomena from nonlinear waves to second-order phase transitions, from superconductivity, superfluidity and Bose-Einstein condensation to liquid crystal and strings in field theory [1]. The existence and uniqueness of weak solution, the regularity and long-time behavior of the solution for the problem can be found in [2][3][4][5]. Numerical methods for Ginzburg-Landau equation include spectral method [6][7][8], finite difference method [9,10], and finite element method (FEM) [11,12] have already been studied. In this article, we are concerned with the following complex Ginzburg-Landau equationu(X, t) = 0, (X, t) ∈ Ω × [0, T], u(X, 0) = u 0 (X), X ∈ Ω,(1.1) Numer Methods Partial Differential Eq. 2020;36:329-341. wileyonlinelibrary.com/journal/num

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Unconditional Superconvergence Analysis for Nonlinear Parabolic Equation with $${\textit{EQ}}_1^{rot}$$ EQ 1 r o t Nonconforming Finite Element

Cited by 72 publications

References 30 publications

Superconvergence analysis of a linearized three‐step backward differential formula finite element method for nonlinear Sobolev equation

Superconvergence analysis of a linearized three‐step backward differential formula finite element method for nonlinear Sobolev equation

A new approach to convergence analysis of linearized finite element method for nonlinear hyperbolic equation

Nonconforming quadrilateral finite element method for Ginzburg–Landau equation

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