Superconvergent analysis of a nonconforming mixed finite element method for time-dependent damped Navier–Stokes equations

Abstract

The main aim of this paper is to develop a nonconforming mixed finite element method for the time-dependent Navier–Stokes problem with nonlinear damping term. The superconvergent analysis of a backward Euler fully-discrete scheme is presented, where the constrained nonconforming rotated (CNR) $$Q_1$$ element and the $$Q_0$$ element are used to approximate the velocity $${\varvec{u}}$$ and the pressure p, respectively. By use of the characters of the element pair together with some striking skills, i.e., mean-value skill and a new transforming skill with respect to $$\tau $$ , the superclose estimates of $$O(h^2+\tau )$$ for $${\varvec{u}}$$ in broken $$H^1$$ -norm and p in $$L^2$$ -norm are deduced rigorously. Furthermore, the global superconvergent results are obtained through the interpolated postprocessing technique. Finally, some numerical results are provided to confirm the theoretical analysis. Here, h is the subdivision parameter and $$\tau $$ , the time step.