Residual-based a posteriori error estimate for interface problems: Nonconforming linear elements

Cai, Zhiqiang; He, Cuiyu; Zhang, Shun

doi:10.1090/mcom/3151

Cited by 14 publications

(28 citation statements)

References 28 publications

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“…The key idea of the proof is to use either element or edge bubble functions in order to localize the error as well as to simplify the boundary conditions. The proof of local efficiency bound similar to (4.1) can be found in [7,4] for the CR nonconforming element.…”

Section: Efficiencymentioning

confidence: 78%

“…This section describes local indicators and global estimators for nonconforming and discontinuous Galerkin finite element approximations. The estimators for the nonconforming elements introduced in [4] and for the discontinuous elements in this paper are more accurate than the existing estimators (see, e.g., [1,5]) and differ in replacing the face tangential derivative jumps by the face solution jumps.…”

Section: Discontinuous Finite Element Approximations and Preliminariesmentioning

confidence: 88%

“…The first two terms of the above equality may be bounded in a similar fashion as that in [4]. That is, it follows from the Cauchy-Schwarz and triangle or trace inequalities, (5.1), and (2.6) that (f, e cr − e cr I ) K ≤ C η cr r,K + osc α (f, K) α 1/2 ∇ h e cr 0,K , ∀ K ∈ T ,…”

Section: Cr Nonconforming Elementsmentioning

confidence: 98%

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Discontinuous Finite Element Methods for Interface Problems: Robust A Priori and A Posteriori Error Estimates

Cai

Zhang³

2017

SIAM J. Numer. Anal.

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For elliptic interface problems, this paper studies residual-based a posteriori error estimations for discontinuous finite element approximations. For the Crouzeix-Raviart nonconforming and the discontinuous Galerkin elements in both two-and three-dimensions, the global reliability bounds are established with constants independent of the jump of the diffusion coefficient. Moreover, we obtain these estimates with no assumption on the distribution of the diffusion coefficient.

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Section: Efficiencymentioning

confidence: 78%

Section: Discontinuous Finite Element Approximations and Preliminariesmentioning

confidence: 88%

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Discontinuous Finite Element Methods for Interface Problems: Robust A Priori and A Posteriori Error Estimates

Cai

Zhang³

2017

SIAM J. Numer. Anal.

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“…For regular facets and elements, there holds the following classical local efficiency results (see [41,19]): The following lemma, which follows from theorem 4.5 in [18], gives the efficiency result for the irregular error terms. Define…”

Section: Efficiencymentioning

confidence: 98%

Flux recovery for Cut Finite Element Method and its application in a posteriori error estimation

Capatina

2021

ESAIM: M2AN

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In this article, we aim to recover locally conservative and $H(div)$ conforming fluxes for the linear Cut Finite Element Solution with Nitsche's method for Poisson problems with Dirichlet boundary condition. The computation of the conservative flux in the Raviart-Thomas space is completely local and does not require to solve any mixed problem. The $L^2$-norm of the difference between the numerical flux and the recovered flux can then be used as a posteriori error estimator in the adaptive mesh refinement procedure. Theoretically we also prove the global reliability and local efficiency. The theoretical results are verified in the numerical results. Moreover, in the numerical results we also observe optimal convergence rate for the flux error.

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“…For regular facets and elements, there holds the following classical local efficiency results (see [40,18]):…”

Section: Efficiencymentioning

confidence: 99%

Flux recovery for Cut finite element method and its application in a posteriori error estimation

Capatina¹,

He²

2021

Preprint

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In this article, we aim to recover locally conservative and H(div) conforming fluxes for the linear Cut Finite Element Solution with Nitsche's method for Poisson problems with Dirichlet boundary condition. The computation of the conservative flux in the Raviart-Thomas space is completely local and does not require to solve any mixed problem. The L 2 -norm of the difference between the numerical flux and the recovered flux can then be used as a posteriori error estimator in the adaptive mesh refinement procedure. Theoretically we are able to prove the global reliability and local efficiency. The theoretical results are verified in the numerical results. Moreover, in the numerical results we also observe optimal convergence rate for the flux error.

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Residual-based a posteriori error estimate for interface problems: Nonconforming linear elements

Cited by 14 publications

References 28 publications

Discontinuous Finite Element Methods for Interface Problems: Robust A Priori and A Posteriori Error Estimates

Discontinuous Finite Element Methods for Interface Problems: Robust A Priori and A Posteriori Error Estimates

Flux recovery for Cut Finite Element Method and its application in a posteriori error estimation

Flux recovery for Cut finite element method and its application in a posteriori error estimation

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