# A POSTERIORI ERROR ESTIMATORS FOR THE STABILIZED LOW-ORDER FINITE ELEMENT DISCRETIZATION OF THE STOKES EQUATIONS BASED ON LOCAL PROBLEMS

• Accepted : 2017.12.11
• Published : 2017.12.25

#### Abstract

In this paper we propose and analyze two a posteriori error estimators for the stabilized $P_1/P_1$ finite element discretization of the Stokes equations. These error estimators are computed by solving local Poisson or Stokes problems on elements of the underlying triangulation. We establish their asymptotic exactness with respect to the velocity error under certain conditions on the triangulation and the regularity of the exact solution.

#### Acknowledgement

Supported by : Kangwon National University

#### References

1. F. Brezzi and J. Pitkaranta, On the stabilization of finite element approximations of the Stokes equations, in Efficient Solutions of Elliptic Systems, Notes on Numerical Fluid Mechanics, Vol. 10, Vieweg+Teubner Verlag, Wiesbaden, 1984.
2. T.J.R. Hughes, L.P. Franca and M. Balestra, A new finite element formulation for computational fluid dynamics: V. Circumventing the Babuska-Brezzi condition: a stable Petrov-Galerkin formulation of the Stokes problem accommodating equal-order interpolations, Computer Methods in Applied Mechanics and Engineering, 59 (1986), 85-99. https://doi.org/10.1016/0045-7825(86)90025-3
3. T.J.R. Hughes and L.P. Franca, A new finite element formulation for computational fluid dynamics: VII. The Stokes problem with various well-posed boundary conditions: Symmetric formulations that converge for all velocity/pressure spaces, Computer Methods in Applied Mechanics and Engineering, 65 (1987), 85-96. https://doi.org/10.1016/0045-7825(87)90184-8
4. F. Brezzi and J. Douglas Jr., Stabilized mixed methods for the Stokes problem, Numerische Mathematik, 53 (1988), 225-235. https://doi.org/10.1007/BF01395886
5. J. Douglas Jr. and J.P.Wang, An absolutely stabilized finite element method for the Stokes problem, Mathematics of Computation, 52 (1989), 495-508. https://doi.org/10.1090/S0025-5718-1989-0958871-X
6. C.R. Dohrmann and P.B. Bochev, A stabilized finite element method for the Stokes problem based on polynomial pressure projections, International Journal for Numerical Methods in Fluids, 46 (2004), 183-201. https://doi.org/10.1002/fld.752
7. P.B. Bochev, C.R. Dohrmann and M.D. Gunzburger, Stabilization of low-order mixed finite elements for the Stokes equations, SIAM Journal on Numerical Analysis, 44 (2006), 82-101. https://doi.org/10.1137/S0036142905444482
8. S. Ganesan, G. Matthies and L. Tobiska, Local projection stabilization of equal order interpolation applied to the Stokes problem, Mathematics of Computation, 77 (2008), 2039-2060. https://doi.org/10.1090/S0025-5718-08-02130-3
9. J. Li and Y. He, A stabilized finite element method based on two local Gauss integrations for the Stokes equations, Journal of Computational and AppliedMathematics, 214 (2008), 58-65.
10. H. Eichel, L. Tobiska and H. Xie, Supercloseness and superconvergence of stabilized loworder finite element discretizations of the Stokes problem, Mathematics of Computation, 80 (2011), 697-722. https://doi.org/10.1090/S0025-5718-2010-02404-4
11. R. Verfurth, A posteriori error estimators for the Stokes equations, Numerische Mathematik, 55 (1989), 309-325. https://doi.org/10.1007/BF01390056
12. R.E. Bank and B.D. Welfert, A posteriori error estimates for the Stokes problem, SIAM Journal on Numerical Analysis, 28 (1991), 591-623. https://doi.org/10.1137/0728033
13. H. Zheng, Y. Hou and F. Shi, A posteriori error estimates of stabilization of low-order mixed finite elements for incompressible flow, SIAM Journal on Scientific Computing, 32 (2010), 1346-1360. https://doi.org/10.1137/090771508
14. L. Song and M. Gao, A posteriori error estimates for the stabilization of low-order mixed finite elements for the Stokes problem, Computer Methods in Applied Mechanics and Engineering, 279 (2014), 410-424. https://doi.org/10.1016/j.cma.2014.07.004
15. J.Wang, Y.Wang and X. Ye, A posteriori error estimate for stabilized finite element methods for the Stokes equations, International Journal of Numerical Analysis and Modeling, 9 (2012), 1-16.
16. R. Stenberg and J. Videman, On the error analysis of stabilized finite element methods for the Stokes problem, SIAM Journal on Numerical Analysis, 53 (2015), 2626-2633. https://doi.org/10.1137/140999396
17. D. Kay and D. Silvester, A posteriori error estimation for stabilized mixed approximations of the Stokes equations, SIAM Journal on Scientific Computing, 21 (1999), 1321-1336. https://doi.org/10.1137/S1064827598333715
18. J. Xu and Z. Zhang, Analysis of recovery type a posteriori error estimators for mildly structured grids, Mathematics of Computation, 73 (2004), 1139-1152.
19. R. Duran and R. Rodriguez, On the asymptotic exactness of Bank-Weiser's estimator, Numerische Mathematik, 62 (1992), 297-303. https://doi.org/10.1007/BF01396231
20. A. Maxim, Asymptotic exactness of an a posteriori error estimator based on the equilibrated residual method, Numerische Mathematik, 106 (2007), 225-253. https://doi.org/10.1007/s00211-007-0064-3