Convergence and Optimality of the Adaptive Nonconforming Linear Element Method for the Stokes Problem

Authors:
Jun Hu;Jinchao Xu
Affiliations:
LMAM, Peking University, Beijing, P.R. China 100871 and School of Mathematical Sciences, Peking University, Beijing, P.R. China 100871;School of Mathematical Sciences, Peking University, Beijing, P.R. China 100871 and Beijing International Center for Mathematical Research, Peking University, Beijing, P.R. China and Department of ...
Venue:
Journal of Scientific Computing
Year:
2013

Citing 24
Cited 2

Error estimators for nonconforming finite element approximations of the Stokes problem

Mathematics of Computation
A convergent adaptive algorithm for Poisson's equation

SIAM Journal on Numerical Analysis
A posteriori error control in low-order finite element discretisations of incompressible stationary flow problems

Mathematics of Computation
Finite Element Method for Elliptic Problems

Finite Element Method for Elliptic Problems
Data Oscillation and Convergence of Adaptive FEM

SIAM Journal on Numerical Analysis
Adaptive Wavelet Methods for Saddle Point Problems---Optimal Convergence Rates

SIAM Journal on Numerical Analysis
An Adaptive Uzawa FEM for the Stokes Problem: Convergence without the Inf-Sup Condition

SIAM Journal on Numerical Analysis
Convergence of Adaptive Finite Element Methods

SIAM Review
Poincaré-Friedrichs Inequalities for Piecewise H1 Functions

SIAM Journal on Numerical Analysis
Local problems on stars: a posteriori error estimators, convergence, and performance

Mathematics of Computation
Adaptive Finite Element Methods with convergence rates

Numerische Mathematik
A unifying theory of a posteriori finite element error control

Numerische Mathematik
Convergence of Adaptive Finite Element Methods for General Second Order Linear Elliptic PDEs

SIAM Journal on Numerical Analysis
Optimality of a Standard Adaptive Finite Element Method

Foundations of Computational Mathematics
A unifying theory of a posteriori error control for nonconforming finite element methods

Numerische Mathematik
Quasi-Optimal Convergence Rate for an Adaptive Finite Element Method

SIAM Journal on Numerical Analysis
An optimally convergent adaptive mixed finite element method

Numerische Mathematik
Convergence of a standard adaptive nonconforming finite element method with optimal complexity

Applied Numerical Mathematics
Convergence and quasi-optimality of adaptive nonconforming finite element methods for some nonsymmetric and indefinite problems

Numerische Mathematik
A Convergent Nonconforming Adaptive Finite Element Method with Quasi-Optimal Complexity

SIAM Journal on Numerical Analysis
Quasi-Optimal Convergence Rate of an Adaptive Discontinuous Galerkin Method

SIAM Journal on Numerical Analysis
Convergence of an Adaptive Mixed Finite Element Method for Kirchhoff Plate Bending Problems

SIAM Journal on Numerical Analysis
Quasi-Optimality of Adaptive Nonconforming Finite Element Methods for the Stokes Equations

SIAM Journal on Numerical Analysis
Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms

Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms

A Mixed Finite Element Method for the Stokes Equations Based on a Weakly Over-Penalized Symmetric Interior Penalty Approach

Journal of Scientific Computing
Axioms of adaptivity

Computers & Mathematics with Applications

Quantified Score

Hi-index	0.00

Visualization

Abstract

In this paper, we analyze the convergence and optimality of a standard adaptive nonconforming linear element method for the Stokes problem. After establishing a special quasi-orthogonality property for both the velocity and the pressure in this saddle point problem, we introduce a new prolongation operator to carry through the discrete reliability analysis for the error estimator. We then use a specially defined interpolation operator to prove that, up to oscillation, the error can be bounded by the approximation error within a properly defined nonlinear approximate class. Finally, by introducing a new parameter-dependent error estimator, we prove the convergence and optimality estimates.