Error analysis of Crouzeix–Raviart and Raviart–Thomas finite element methods

Kobayashi, Kensuke; Tsuchiya, Takuya

doi:10.1007/s13160-018-0325-9

Cited by 6 publications

(3 citation statements)

References 15 publications

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“…However, the maximum-angle condition is not necessarily needed to obtain error estimates. Recently, in the two-dimensional instance, the CR finite-element analysis of the non-homogeneous Dirichlet-Poisson problem has been investigated under a more relaxed mesh condition, [18]. The present paper extends previous research to a three-dimensional setting.…”

Section: Introductionmentioning

confidence: 59%

Crouzeix-Raviart and Raviart-Thomas finite-element error analysis on anisotropic meshes violating the maximum-angle condition

Ishizaka

Kobayashi

Tsuchiya

2020

Preprint

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We investigate the piecewise linear nonconforming Crouzeix-Raviar and the lowest order Raviart-Thomas finite-element methods for the Poisson problem on three-dimensional anisotropic meshes. We first give error estimates of the Crouzeix-Raviart and the Raviart-Thomas finite-element approximate problems. We next present the equivalence between the Raviart-Thomas finiteelement method and the enriched Crouzeix-Raviart finite-element method. We emphasise that we do not impose either shape-regular or maximum-angle condition during mesh partitioning. Numerical results confirm the results that we obtained.

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Section: Introductionmentioning

confidence: 59%

Crouzeix-Raviart and Raviart-Thomas finite-element error analysis on anisotropic meshes violating the maximum-angle condition

Ishizaka

Kobayashi

Tsuchiya

2020

Preprint

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“…where l 1 ≤ l 2 ≤ • • • ≤ h T are the lengths of the edges of T . As has been seen in [8,9,10,11,12,13], R T is an important parameter in measuring interpolation errors on d-simplices. For example, the errors of Lagrange interpolation on T are bounded in terms of R T , as presented by (5.10) and (5.11).…”

Section: Meshes Of ωmentioning

confidence: 99%

A robust discontinuous Galerkin scheme on anisotropic meshes

Kashiwabara,

Tsuchiya

2020

Preprint

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Discontinuous Galerkin (DG) methods are extensions of the usual Galerkin finite element methods. Although there are vast amount of studies on DG methods, most of them have assumed shape-regularity conditions on meshes for both theoretical error analysis and practical computations. In this paper, we present a new symmetric interior penalty DG scheme with a modified penalty term. We show that, without imposing the shape-regularity condition on the meshes, the new DG scheme inherits all of the good properties of standard DG methods, and is thus robust on anisotropic meshes. Numerical experiments confirm the theoretical error estimates obtained.

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“…Meanwhile, in [16], the lowest-order Raviart-Thomas interpolation error analysis under a condition weaker than the maximum-angle condition was introduced in the two-dimensional case. The analysis was based on the technique of Babuska and Aziz [4].…”

Section: Introductionmentioning

confidence: 99%

General theory of interpolation error estimates on anisotropic meshes

Ishizaka

Kobayashi

Tsuchiya

2020

Preprint

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We propose a general theory of estimating interpolation error for smooth functions in two and three dimensions. In our theory, the error of interpolation is bound in terms of the diameter of a simplex and a geometric parameter. In the two-dimensional case, our geometric parameter is equivalent to the circumradius of a triangle. In the three-dimensional case, our geometric parameter also represents the flatness of a tetrahedron. Through the introduction of the geometric parameter, the error estimates newly obtained can be applied to cases that violate the maximum-angle condition.

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Error analysis of Crouzeix–Raviart and Raviart–Thomas finite element methods

Cited by 6 publications

References 15 publications

Crouzeix-Raviart and Raviart-Thomas finite-element error analysis on anisotropic meshes violating the maximum-angle condition

Crouzeix-Raviart and Raviart-Thomas finite-element error analysis on anisotropic meshes violating the maximum-angle condition

A robust discontinuous Galerkin scheme on anisotropic meshes

General theory of interpolation error estimates on anisotropic meshes

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