Fully Discrete Approximations to the Time-Dependent Navier–Stokes Equations with a Projection Method in Time and Grad-Div Stabilization

Abstract

This paper studies fully discrete approximations to the evolutionary Navier–Stokes equations by means of inf-sup stable $$H^1$$ -conforming mixed finite elements with a grad-div type stabilization and the Euler incremental projection method in time. We get error bounds where the constants do not depend on negative powers of the viscosity. We get the optimal rate of convergence in time of the projection method. For the spatial error we get a bound $$O(h^k)$$ for the $$L^2$$ error of the velocity, k being the degree of the polynomials in the velocity approximation. We prove numerically that this bound is sharp for this method.