Residual and hierarchical<i><b>a</b>posteriori</i>error estimates for nonconforming mixed finite element methods

Alaoui, Linda El; Ern, Alexandre

doi:10.1051/m2an:2004044

Cited by 19 publications

(23 citation statements)

References 33 publications

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“…The operator I Os is sometimes referred to as the Oswald interpolation operator; it has been considered in [5,13,16,22]. Another hp-interpolation operator for non-smooth functions generalizing that of Clément and Scott-Zhang is analyzed in [23].…”

Section: Continuous Hp-interpolation the Goal Is To Construct An Opementioning

confidence: 99%

Continuous interior penalty $hp$-finite element methods for advection and advection-diffusion equations

2007

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Abstract. A continuous interior penalty hp-finite element method that penalizes the jump of the gradient of the discrete solution across mesh interfaces is introduced. Error estimates are obtained for advection and advection-diffusion equations. The analysis relies on three technical results that are of independent interest: an hp-inverse trace inequality, a local discontinuous to continuous hp-interpolation result, and hp-error estimates for continuous L 2 -orthogonal projections.

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Section: Continuous Hp-interpolation the Goal Is To Construct An Opementioning

confidence: 99%

Continuous interior penalty $hp$-finite element methods for advection and advection-diffusion equations

2007

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“…) be the so-called Oswald interpolation operator (El Alaoui & Ern, 2004;Hoppe & Wohlmuth, 1996) defined as follows:…”

Section: The Discrete Settingmentioning

confidence: 99%

A priori and a posteriori analysis of non-conforming finite elements with face penalty for advection–diffusion equations

Alaoui¹,

Ern²,

Burman³

2007

IMA Journal of Numerical Analysis

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We analyse a non-conforming finite-element method to approximate advection-diffusion-reaction equations. The method is stabilized by penalizing the jumps of the solution and those of its advective derivative across mesh interfaces. The a priori error analysis leads to (quasi-)optimal estimates in the mesh size (sub-optimal by order 1 2 in the L 2 -norm and optimal in the broken graph norm for quasi-uniform meshes) keeping the Péclet number fixed. Then, we investigate a residual a posteriori error estimator for the method. The estimator is semi-robust in the sense that it yields lower and upper bounds of the error which differ by a factor equal at most to the square root of the Péclet number. Finally, to illustrate the theory we present numerical results including adaptively generated meshes.

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“…The technique consists in projecting the FVE space to another approximation space (possibly of higher order) related to a coarser mesh. A detailed study including the analysis of a posteriori error estimates for FVE methods in the spirit of [5,12], and adaptivity following [8] have been postponed for a forthcoming paper. Further efforts are also being made to extend the analysis herein presented to the transient Navier-Stokes equations.…”

Section: Introductionmentioning

confidence: 99%

Analysis of a finite volume element method for the Stokes problem

Quarteroni

Ruiz–Baier

2011

Numer. Math.

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Abstract. In this paper we propose a stabilized conforming finite volume element method for the Stokes equations. On stating the convergence of the method, optimal a priori error estimates in different norms are obtained by establishing the adequate connection between the finite volume and stabilized finite element formulations. A superconvergence result is also derived by using a postprocessing projection method. The stabilization of the continuous lowest equal order pair finite volume element discretization (P 1 − P 1 ) is achieved by enriching the velocity space with bubble-like functions. Finally, some numerical experiments that confirm the predicted behavior of the method are provided.

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Residual and hierarchicalaposteriorierror estimates for nonconforming mixed finite element methods

Cited by 19 publications

References 33 publications

Continuous interior penalty $hp$-finite element methods for advection and advection-diffusion equations

Continuous interior penalty $hp$-finite element methods for advection and advection-diffusion equations

A priori and a posteriori analysis of non-conforming finite elements with face penalty for advection–diffusion equations

Analysis of a finite volume element method for the Stokes problem

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