Convergence and superconvergence of a nonconforming finite element method for the Stokes problem

In this paper, the four-parameter nonconforming finite element proposed in [30] and [19] is analyzed with the framework of Double Set Parameter (DSP) method, then it is applied to the stationary Stokes problem. The element exhibits some features of the well-known Q 1−P 0 element under rectangular meshes. An optimal convergence rate is established for both the velocity and smoothed pressure. Furthermore, the superconvergent approximation between the interpolation of the exact solution and the finite element solution is proved. A superconvergent estimate on the centers of elements and the global superconvergence for the gradient of the velocity and the pressure are derived with the aid of a postprocessing method. Based on the superconvergence property, an asymptotically exact a posteriori estimator of ZZ type is also studied.