Journal of Scientific Computing

, Volume 55, Issue 1, pp 125–148 | Cite as

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

  • Jun Hu
  • Jinchao Xu


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.


Adaptive finite element method Convergence Optimality The Stokes problem 



The first author was supported by NSFC 10971005, and in part by NSFC 11031006. The second author was supported in part, by NSFC-10528102, NSF DMS 0915153, and DMS 0749202, and by the PSU-PKU Joint Center for Computational Mathematics and Applications.


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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.LMAMPeking UniversityBeijingP.R. China
  2. 2.School of Mathematical SciencesPeking UniversityBeijingP.R. China
  3. 3.Beijing International Center for Mathematical ResearchPeking UniversityBeijingP.R. China
  4. 4.Department of MathematicsPennsylvania State UniversityUniversity ParkUSA

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