Forty Years of the Crouzeix‐Raviart element

Brenner, Susanne C.

doi:10.1002/num.21892

Cited by 65 publications

(44 citation statements)

References 151 publications

(168 reference statements)

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“…where we use the discrete Sobolev inequality and the fact that ∇χ belongs to the Crouzeix-Raviar finite element space [9]. This implies the bound (3.19).…”

Section: Fully Discrete Approximationmentioning

confidence: 98%

Analysis of the Morley element for the Cahn–Hilliard equation and the Hele-Shaw flow

2020

ESAIM: M2AN

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The paper analyzes the Morley element method for the Cahn-Hilliard equation. The objective is to derive the optimal error estimates and to prove the zero-level sets of the Cahn-Hilliard equation approximate the Hele-Shaw flow. If the piecewise L ∞ (H 2 ) error bound is derived by choosing test function directly, we cannot obtain the optimal error order, and we cannot establish the error bound which depends on 1 polynomially either. To overcome this difficulty, this paper proves them by the following steps, and the result in each next step cannot be established without using the result in its previous one. First, it proves some a priori estimates of the exact solution u, and these regularity results are minimal to get the main results; Second, it establishes L ∞ (L 2 ) and piecewise L 2 (H 2 ) error bounds which depend on 1 polynomially based on the piecewise L ∞ (H −1 ) and L 2 (H 1 ) error bounds; Third, it establishes piecewise L ∞ (H 2 ) optimal error bound which depends on 1 polynomially based on the piecewise L ∞ (L 2 ) and L 2 (H 2 ) error bounds; Finally, it proves the L ∞ (L ∞ ) error bound and the approximation to the Hele-Shaw flow based on the piecewise L ∞ (H 2 ) error bound. The nonstandard techniques are used in these steps such as the generalized coercivity result, integration by part in space, summation by part in time, and special properties of the Morley elements. If one of these techniques is lacked, either we can only obtain the sub-optimal piecewise L ∞ (H 2 ) error order, or we can merely obtain the error bounds which are exponentially dependent on 1 . The approach used in this paper provides a way to bound the errors in higher norm from the errors in lower norm step by step, which has a profound meaning in methodology. Numerical results are presented to validate the optimal L ∞ (H 2 ) error order and the asymptotic behavior of the solutions of the Cahn-Hilliard equation. m 0 := 1 |Ω| Ω u 0 (x) dx.(2) There exists a nonnegative constant σ 1 such that

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“…where we use the discrete Sobolev inequality and the fact that ∇χ belongs to the Crouzeix-Raviar finite element space [9]. This implies the bound (3.19).…”

Section: Fully Discrete Approximationmentioning

confidence: 98%

Analysis of the Morley element for the Cahn–Hilliard equation and the Hele-Shaw flow

2020

ESAIM: M2AN

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“…As a direct application of the gradient recovery method, we naturally define a recovery-type a posteriori error estimator for the surface Crouzeix-Raviart element. The local a posteriori error estimator on each element T is defined as 9) and the global error estimator as…”

Section: )mentioning

confidence: 99%

Surface Crouzeix–Raviart element for the Laplace–Beltrami equation

Guo

2020

Numer. Math.

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This paper is concerned with the nonconforming finite element discretization of geometric partial differential equations. In specific, we construct a surface Crouzeix-Raviart element on the linear approximated surface, analogous to a flat surface. The optimal error estimations are established even though the presentation of the geometric error. By taking the intrinsic viewpoint of manifolds, we introduce a new superconvergent gradient recovery method for the surface Crouzeix-Raviart element using only the information of discretization surface. The potential of serving as an asymptotically exact a posteriori error estimator is also exploited. A series of benchmark numerical examples are presented to validate the theoretical results and numerically demonstrate the superconvergence of the gradient recovery method.

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“…In some cases, the condition (H2) is a strong assumption. For example, generally the shape functions of the Crouzeix-Raviart non-conforming element [11] are only continuous at n − 1 points on a n − 1 dimensional face. However, with the DoF-transfer operation, the resulting finite element functions are continuous at each vertex.…”

Section: A Discrete Korn Inequalitymentioning

confidence: 99%

Nonstandard finite element de Rham complexes on cubical meshes

Gillette

Zhang

2019

Bit Numer Math

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We propose two general operations on finite element differential complexes on cubical meshes that can be used to construct and analyze sequences of "nonstandard" convergent methods. The first operation, called DoF-transfer, moves edge degrees of freedom to vertices in a way that reduces global degrees of freedom while increasing continuity order at vertices. The second operation, called serendipity, eliminates interior bubble functions and degrees of freedom locally on each element without affecting edge degrees of freedom. These operations can be used independently or in tandem to create nonstandard complexes that incorporate Hermite, Adini and "trimmed-Adini" elements. We show that the resulting elements provide convergent, non-conforming methods for problems requiring stronger regularity and satisfy a discrete Korn inequality. We discuss potential benefits of applying these elements to Stokes, biharmonic and elasticity problems.

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Forty Years of the Crouzeix‐Raviart element

Cited by 65 publications

References 151 publications

Analysis of the Morley element for the Cahn–Hilliard equation and the Hele-Shaw flow

Analysis of the Morley element for the Cahn–Hilliard equation and the Hele-Shaw flow

Surface Crouzeix–Raviart element for the Laplace–Beltrami equation

Nonstandard finite element de Rham complexes on cubical meshes

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