A posteriori error estimators for stabilized P1 nonconforming approximation of the Stokes problem

In this paper we derive and analyze some a posteriori error estimators for the stabilized P1 nonconforming approximation of the Stokes problem involving the strain tensor. This will be done by decomposing the numerical error in a proper way into conforming and nonconforming contributions. The error estimator for the nonconforming error is obtained in the standard way, and the implicit error estimator for the conforming error is derived by applying the equilibrated residual method. A crucial part of this work is construction of approximate normal stresses on interelement boundaries which will serve as equilibrated Neumann data for local Stokes problems. It turns out that such normal stresses can be simply computed by local weak residuals of the discrete system plus jumps of the velocity solution and that a stronger equilibration condition is satisfied to ensure solvability of local Stokes problems. We also derive a simple explicit error estimator based on the nonsymmetric tensor recovery of the normal stress error. Numerical results are provided to illustrate the performance of our error estimators.