Journal of Scientific Computing

, Volume 55, Issue 1, pp 125–148

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

• Jun Hu
• Jinchao Xu
Article

## 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.

## Keywords

Adaptive finite element method Convergence Optimality The Stokes problem

## Notes

### Acknowledgements

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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