Journal ArticleDOI
Error estimators for nonconforming finite element approximations of the Stokes problem
TLDR
This paper defines and analyze a posteriori error estimators for nonconforming approximations of the Stokes equations and proves that these estimators are equivalent to an appropriate norm of the error.Abstract:
In this paper we define and analyze a posteriori error estimators for nonconforming approximations of the Stokes equations. We prove that these estimators are equivalent to an appropriate norm of the error. For the case of piecewise linear elements we define two estimators. Both of them are easy to compute, but the second is simpler because it can be computed using only the right-hand side and the approximate velocity. We show how the first estimator can be generalized to higher-order elements. Finally, we present several numerical examples in which one of our estimators is used for adaptive refinement.read more
Citations
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Journal ArticleDOI
Axioms of adaptivity
TL;DR: In this paper, efficiency exclusively characterizes the approximation classes involved in terms of the best-approximation error and data resolution and so the upper bound on the optimal marking parameters does not depend on the efficiency constant.
Journal ArticleDOI
Robust A Posteriori Error Estimation for Nonconforming Finite Element Approximation
TL;DR: The equilibrated residual method for a posteriori error estimation is extended to nonconforming finite element schemes for the approximation of linear second order elliptic equations where the permeability coefficient is allowed to undergo large jumps in value across interfaces between differing media.
Journal ArticleDOI
A unifying theory of a posteriori error control for nonconforming finite element methods
Carsten Carstensen,Jun Hu +1 more
TL;DR: In this paper, a posteriori error estimation for nonconforming finite element methods (NCFEMs) on parallelograms has been derived for the Laplace, Stokes, and Navier-Lame equations.
Journal ArticleDOI
A unifying theory of a posteriori finite element error control
TL;DR: Residual-based a posteriori error estimates are derived within a unified setting for lowest-order conforming, nonconforming, and mixed finite element schemes using the operator norm ||ℓ|| of a linear functional of the form in the variable υ of a Sobolev space V.
Journal ArticleDOI
A posteriori error control in low-order finite element discretisations of incompressible stationary flow problems
TL;DR: Computable a posteriori error bounds and related adaptive meshrefining algorithms are provided for the numerical treatment of monotone stationary flow problems with a quite general class of conforming and non-conforming finite element methods.
References
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Book
Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms
TL;DR: This paper presents the results of an analysis of the "Stream Function-Vorticity-Pressure" Method for the Stokes Problem in Two Dimensions and its applications to Mixed Approximation and Homogeneous Stokes Equations.
Journal ArticleDOI
Finite element interpolation of nonsmooth functions satisfying boundary conditions
L. Ridgway Scott,Shangyou Zhang +1 more
TL;DR: In this article, a modified Lagrange type interpolation operator is proposed to approximate functions in Sobolev spaces by continuous piecewise polynomials, and the combination of averaging and interpolation is shown to be a projection, and optimal error estimates are proved for the projection error.
Journal ArticleDOI
Conforming and nonconforming finite element methods for solving the stationary Stokes equations I
M. Crouzeix,P.-A. Raviart +1 more
TL;DR: Both conforming and nonconforming finite element methods are studied and various examples of simplicial éléments well suited for the numerical treatment of the incompressibility condition are given.
Journal ArticleDOI
A‐posteriori error estimates for the finite element method
Ivo Babuška,Werner C. Rheinboldt +1 more
TL;DR: In this article, a-posteriori error estimates for finite element solutions are derived in an asymptotic form for h 0 where h measures the size of the elements.