Fully computable a posteriori error estimates for the Stokes equation without the global inf–sup constant

In this paper we consider the a posteriori error estimates for the Stokes equation which provide computable upper bounds on the actual errors. It is known that such error estimates typically involve the global inf–sup constant which can be quite tricky to find and lead to extra difficulty in numerical computations. To resolve this difficulty, we propose a new error estimate which relies only upon the inf–sup constants local to the subdomains forming a partition of the original domain. Hence our new error estimate is fully computable whenever the subdomains are simple enough to make the local inf–sup constants readily available, and moreover, provides a sharper upper bound than the previous estimate when these local constants are bigger than the global constant. Application to the Crouzeix–Raviart and Fortin–Soulie nonconforming finite elements is presented along with some effective minimization technique for further improvement of the upper bound on the error. Finally, numerical experiments are carried out to investigate the performance of the new error estimate.