Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-10T16:15:54.807Z Has data issue: false hasContentIssue false
  • English
  • Français

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)
Get access

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Bank, R. E. and Welfert, B. D., A posteriori error estimates for the Stokes equations: A comparison, Comp. Meth. Appl. Mech. Engg. 82, 323–340 (1990).CrossRefGoogle Scholar
[2] Bank, R. E. and Welfert, B. D., A posteriori error estimates for the Stokes problem, SIAM J. Num. An. 28, 591–623 (1991).Google Scholar
[3] Carstensen, C., Causin, P. and Sacco, R., A posteriori dual-mixed adaptive finite element error control for Lamand Stokes equations, Num. Mathematik 101, 309–332 (2005).Google Scholar
[4] Carstensen, C. and Funken, S., A posteriori error control in low-order finite element discretisations of incompressible stationary flow problems, Math. Comp. 70, 1353–1381 (2001).Google Scholar
[5] Chen, G., Feng, M. and Xie, X., Robust globally divergence-free weak Galerkin methods for Stokes equations, J. Comput. Math. 34, 549–572 (2016).Google Scholar
[6] Chen, L., Wang, J. and Ye, X., A posteriori error estimates for weak Galerkin finite element methods for second order elliptic problems, J. Scientific Comp. 59, 496–511 (2014).Google Scholar
[7] Congreve, S., Houston, P., Suli, E. and Wihler, T.P., Discontinuous Galerkin finite element approximation of quasilinear elliptic boundary value problems II: Strongly monotone quasi-Newtonian flows, IMA J. Num. Analysis 33, 1386–1415 (2013).Google Scholar
[8] Dari, E., Durán, E. and Padra, C., Error estimators for nonconforming finite element approximations of the Stokes problem, Math. Comp. 64, 1017–1033 (1995).Google Scholar
[9] Dörfler, W. and Ainsworth, M., Reliable a posteriori error control for nonconforming finite element approximation of Stokes flow, Math. Comp. 74 1599–1619 (2005).Google Scholar
[10] Farhloul, M., Nicaise, S. and Paque L, L., A posteriori error estimation for the dual mixed finite element method of the Stokes problem, C. R. Acad. Sci. Paris, Ser. I. 339, 513–518 (2004).CrossRefGoogle Scholar
[11] Hannukainen, A., Stenberg, R. and VohralŠk, M., A unified framework for a posteriori error estimation for the Stokes problem, Num. Mathematik 122, 725–769 (2012).Google Scholar
[12] Kay, D. and Silvester, D., A posteriori error estimation for stabilized mixed of the Stokes equations, SIAM J. Scientific Comp. 21, 1321–1336 (1999).Google Scholar
[13] Hosking, R.J. and Dewar, R.L., Fundamental Fluid Mechanics and Magnetohydrodynamics, Springer (2016).Google Scholar
[14] Houston, P., Schotzau, D. and Wihler, T.P., hp-adaptive discontinuous Galerkin finite element methods for the Stokes proble, European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2004), Eds. Neittaanmaki, P., Rossi, P., Korotov, S., Onate, E., Periaux, J. and Knorzer, D., Jyvaskyla, 24-28 July 2004 Google Scholar
[15] Mitchell, W.F., A comparison of adaptive refinement techniques for elliptic problems, ACMTrans. Math. Software (TOMS) 15, 326–347 (1989).Google Scholar
[16] Paul, H., Schözau, D. and Wihler, T. P., Energy norm shape a posteriori error estimation for mixed discontinuous Galerkin approximations of the Stokes problem, J. Scientific Comp. 22, 347–370 (2005).Google Scholar
[17] Shi, Z. and Wang, M., Finite Element Methods. Science Press (2013).Google Scholar
[18] Verfürth, R., A posteriori error estimators for the Stokes equations, J. Num. Mathematik, 55, 309–325 (1989).Google Scholar
[19] Verfürth, R., A posteriori error estimators for the Stokes equations II non-conforming discretizations, J. Num. Mathematik 60, 235–249 (1991).Google Scholar
[20] Verfürth, R., A Review of A posteriori Error Estimation and Adaptive Mesh Refinement Techniques, Wiley Teubner, Chichester and New York (1996).Google Scholar
[21] Wang, J. and Ye, X., A weak Galerkin finite element method for the Stokes equations, Adv. Comp. Math. 42, 1–20 (2015).Google Scholar
[22] Wang, R., Wang, X., Zhai, Q. and Zhang, R., A Weak Galerkin Finite Element Scheme for solving the stationary Stokes equations, J. Comp. Appl. Math. 302, 171–185 (2016).Google Scholar
[23] Zheng, X., Chen, G. and Xie, X., A divergence-free weak Galerkin method for quasi-Newtonian Stokes flows, Science China Math. 60, doi: 10.1007/s11425-016-0354-8 (2017).Google Scholar