Finite element interpolation of nonsmooth functions satisfying boundary conditions

In this paper, we derive quasi-norm a priori and a posteriori error estimates for the Crouzeix-Raviart type finite element approximation of the p-Laplacian. Sharper a priori upper error bounds are obtained. For instance, for sufficiently regular solutions we prove optimal a priori error bounds on the discretization error in an energy norm when 1 < p ≤ 2. We also show that the new a posteriori error estimates provide improved upper and lower bounds on the discretization error. For sufficiently regular solutions, the a posteriori error estimates are further shown to be equivalent on the discretization error in a quasi-norm. Classification (1991): 65N30, 49J40 Mathematics Subject

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“…If one uses a standard averaging interpolator such as the one introduced in [25], then one has to estimate the integrals like…”

Section: It Follows From Proposition 31 Thatmentioning

confidence: 99%

Quasi-norm a priori and a posteriori error estimates for the nonconforming approximation of p-Laplacian

Liu

Yan

2001

Numer. Math.

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“…The usual Lagrange interpolation operator is not stable in H 1 and the averaging type operators of Clément [8] and Scott-Zhang [21] do not satisfy (1.7). In Sect.…”

Section: Introductionmentioning

confidence: 99%

Residual type a posteriori error estimates for elliptic obstacle problems

Chen¹,

Nochetto

2000

Numerische Mathematik

164

161

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A posteriori error estimators of residual type are derived for piecewise linear finite element approximations to elliptic obstacle problems. An instrumental ingredient is a new interpolation operator which requires minimal regularity, exhibits optimal approximation properties and preserves positivity. Both upper and lower bounds are proved and their optimality is explored with several examples. Sharp a priori bounds for the a posteriori estimators are given, and extensions of the results to double obstacle problems are briefly discussed.Mathematics Subject Classification (1991): 65N15, 65N30; 41A05, 41A29, 41A36

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“…These can be obtained by regularization, an approach that is similar to that used by Brenner in [8]. The following simplified variant of the Scott and Zhang regularization [34] extends to three dimensions an idea of M. Crouzeix [11]. Let P h denote the set of vertices of T h .…”

Section: Properties Of M H and Discrete Inf-sup Conditionmentioning

confidence: 99%

Numerical discretization of a Darcy–Forchheimer model

Girault

Wheeler

2008

Numer. Math.

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We solve a steady Darcy-Forchheimer flow in a bounded region by means of piecewise constant velocities and nonconforming piecewise P 1 pressures. For the computation, we solve the nonlinearity by an alternating-directions algorithm and we decouple the computation of the velocity from that of the pressure by a gradient algorithm. We prove a priori error estimates of the scheme and convergence of the alternating-directions algorithm.

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Finite element interpolation of nonsmooth functions satisfying boundary conditions

Cited by 1,220 publications

References 3 publications

Quasi-norm a priori and a posteriori error estimates for the nonconforming approximation of p-Laplacian

Quasi-norm a priori and a posteriori error estimates for the nonconforming approximation of p-Laplacian

Residual type a posteriori error estimates for elliptic obstacle problems

Numerical discretization of a Darcy–Forchheimer model

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