Superconvergence Analysis of a Nonconforming Triangular Element on Anisotropic Meshes

Shi, Dongyang; Liang, Hui; Wang, Caixia

doi:10.1007/s11424-007-9051-0

Cited by 10 publications

(11 citation statements)

References 14 publications

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“…Similarly, Next, putting the given results into (25), integrating with respect to t and noting that t .0/ D 0, Á.0/ D 0, then by applying Gronwall's lemma, we can obtain that…”

Section: Proofmentioning

confidence: 95%

New estimates of mixed finite element method for fourth‐order wave equation

Shi

Wang

Liao

2017

Math Methods in App Sciences

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Communicated by S. WiseThe conforming bilinear mixed finite element method is established for a fourth-order wave equation. By applying the interpolation and projection simultaneously, the superclose and superconvergence results of order O.h 2 / for the original variable u and intermediate variable p D a 1 .X/u in H 1 -norm are achieved for the semi-discrete and fully discrete schemes, respectively (where a 1 .X/ is the coefficient appeared in the equation). The distinct advantage of this method is that it can reduce the smoothness of solutions u, u t , and p compared with the existing literature for getting optimal estimates alone. At last, some numerical results are carried out to validate the theoretical analysis. 4448D. SHI ET AL. projection. Secondly, through establishing the relationship between the Ritz projection and interpolation of the bilinear element, the superclose estimate of the interpolation for u in H 1 -norm is obtained. Thirdly, the superclose property of the interpolation for p in H 1norm is obtained under the condition a 1 .X/ D 1 c.X/ . Then the global superconvergence results for the given two variables are derived with the help of the interpolation post-processing technique. At last, some numerical results are provided to illustrate the validity of the theoretical analysis and effectiveness of the proposed method.It should be pointed out that the superclose result of order O.h 2 / for u in H 1 -norm and convergence result for p in L 2 -norm are obtained by use of the interpolation operator alone in [6], the solutions u, u t , and p are required belonging to H 4 . /. However, in this paper, by applying the interpolation and projection simultaneously, we prove the superclose results of order O.h 2 / for both u and p in H 1 -norm under weaker assumptions u, u t , p 2 H 3 . /. On the other hand, if the projection is utilized alone, we cannot derive the global superconvergence results for how to construct a suitable post-processing operator for the projection still remains open.The outline of this paper is organized as follows: In Section 2, the MFE spaces and a lemma are introduced. In Sections 3 and 4, the superclose and superconvergence results are proved for the semi-discrete and fully discrete schemes, respectively. In Section 5, some remarks are given. In the last section, two numerical examples are presented.Throughout this article, C denotes a positive constant that may take different values at different places but remains independent of the mesh parameter h.C .a 2 .X/rÂ, r /. Remark 4From the proofs of Theorems 3.3 and 4.3, we can find that the constraints on a 2 .X/ and a 3 .X/ are stronger than that in the paper [6], and we derive the superclose property of p under the condition c.X/ D b.X/; how to get rid of this restriction and extend to the general situation is one of topics in our further work. Numerical resultsExample 6.1 Consider the following constant coefficient equation (a 1 .X/ D a 2 .X/ D a 3 .X/ D c.X/ D 1) u tt C 2 u u C u D f .X, t/, in .0, T/, @u @n D @.u/ @n D 0, on @...

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“…Similarly, Next, putting the given results into (25), integrating with respect to t and noting that t .0/ D 0, Á.0/ D 0, then by applying Gronwall's lemma, we can obtain that…”

Section: Proofmentioning

confidence: 95%

New estimates of mixed finite element method for fourth‐order wave equation

Shi

Wang

Liao

2017

Math Methods in App Sciences

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“…In the past, the H 1 ‐superconvergence results for the FE method for the 2D Poisson equation were studied by Chen and Huang , Chen , Lin and Yan , Shi and Liang , and Zhu and Lin . In this paper, we consider the FD method for the 3D Poisson equation by using the Q 1 ‐conforming element on a quasi‐uniform mesh.…”

Section: Introductionmentioning

confidence: 99%

H¹‐superconvergence of finite difference method based on Q₁‐element on quasi‐uniform mesh for the 3D Poisson equation

Feng

Chen

2019

Numerical Methods Partial

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In this paper, the finite difference (FD) method is considered for the 3D Poisson equation by using the Q1‐element on a quasi‐uniform mesh. First, under the regularity assumption of u∈H3Ω∩H01Ω, the H1‐superconvergence of the FD solution uh based on the Q1‐element to the first‐order interpolation function Ih1u is obtained. Next, the H1‐superconvergence of the second‐order interpolation postprocessing function I2h2uh based on the FD solution uh to u is provided. Finally, numerical tests are presented to show the H1‐superconvergence result of the FD postprocessing function I2h2uh to u if u∈H3Ω∩H01Ω.

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“…[19][20][21][22][23]). However, the applications of Crouzeix-Raviart type anisotropic nonconforming linear triangular element [14] and EQ rot 1 rectangular element [24] to the variational inequality problem (1.1) have never been seen, although [25] applied the latter one to a class of nonlinear Sobolev equations and obtained the optimal error estimates and the supercloseness for both semi-discrete and fullydiscrete approximate schemes, and [26] discussed the supercloseness and superconvergence of nonconforming rectangular elements for the second order elliptic problems.…”

Section: Introductionmentioning

confidence: 99%

Anisotropic Crouzeix-Raviart Type Nonconforming Finite Element Methods to Variational Inequality Problem with Displacement Obstacle

Shi¹

2015

JCM

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In this paper, anisotropic Crouzeix-Raviart type nonconforming finite element methods are considered for solving the second order variational inequality with displacement obstacle. The convergence analysis is presented and the optimal order error estimates are obtained under the hypothesis of the finite length of the free boundary. Numerical results are provided to illustrate the correctness of theoretical analysis.Mathematics subject classification: 65N15, 65N30.

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Superconvergence Analysis of a Nonconforming Triangular Element on Anisotropic Meshes

Cited by 10 publications

References 14 publications

New estimates of mixed finite element method for fourth‐order wave equation

New estimates of mixed finite element method for fourth‐order wave equation

H¹‐superconvergence of finite difference method based on Q₁‐element on quasi‐uniform mesh for the 3D Poisson equation

Anisotropic Crouzeix-Raviart Type Nonconforming Finite Element Methods to Variational Inequality Problem with Displacement Obstacle

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Superconvergence Analysis of a Nonconforming Triangular Element on Anisotropic Meshes

Cited by 10 publications

References 14 publications

New estimates of mixed finite element method for fourth‐order wave equation

New estimates of mixed finite element method for fourth‐order wave equation

H1‐superconvergence of finite difference method based on Q1‐element on quasi‐uniform mesh for the 3D Poisson equation

Anisotropic Crouzeix-Raviart Type Nonconforming Finite Element Methods to Variational Inequality Problem with Displacement Obstacle

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H¹‐superconvergence of finite difference method based on Q₁‐element on quasi‐uniform mesh for the 3D Poisson equation