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A Posteriori Error Estimator for a Weak Galerkin Finite Element Solution of the Stokes Problem

Part of: Incompressible viscous fluids Partial differential equations, boundary value problems

Published online by Cambridge University Press:  07 September 2017

Xiaobo Zheng  and
Xiaoping Xie
Xiaobo Zheng*
Affiliation:
School of Mathematics, Sichuan University, Chengdu 610064, China
Xiaoping Xie*
Affiliation:
School of Mathematics, Sichuan University, Chengdu 610064, China
*
*Corresponding author. Email addresses:zhengxiaobosc@yahoo.com (X. Zheng), xpxie@scu.edu.cn (X. Xie)
*Corresponding author. Email addresses:zhengxiaobosc@yahoo.com (X. Zheng), xpxie@scu.edu.cn (X. Xie)
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Abstract

A robust residual-based a posteriori error estimator is proposed for a weak Galerkin finite element method for the Stokes problem in two and three dimensions. The estimator consists of two terms, where the first term characterises the difference between the L2-projection of the velocity approximation on the element interfaces and the corresponding numerical trace, and the second is related to the jump of the velocity approximation between the adjacent elements. We show that the estimator is reliable and efficient through two estimates of global upper and global lower bounds, up to two data oscillation terms caused by the source term and the nonhomogeneous Dirichlet boundary condition. The estimator is also robust in the sense that the constant factors in the upper and lower bounds are independent of the viscosity coefficient. Numerical results are provided to verify the theoretical results.

Keywords

The Stokes equationsweak Galerkin methoda posteriori error estimator

MSC classification

Secondary: 65N15: Error bounds 65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 76D07: Stokes and related (Oseen, etc.) flows
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 7 , Issue 3 , August 2017 , pp. 508 - 529
Copyright
Copyright © Global-Science Press 2017 

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