A posteriori error estimations for mixed finite element approximations to the Navier–Stokes equations based on Newton-type linearization

Abstract

In this paper we derive a posteriori error estimates for inf–sup stable mixed finite element approximations to the evolutionary Navier–Stokes equations. We reduce the problem of getting a posteriori error estimations of a non-linear evolutionary problem to that of getting a posteriori estimations of a linear steady problem. The main idea is based on the fact that the solutions of some Newton-type linearized Navier–Stokes equations around the plain Galerkin approximations approach the solution of the original Navier–Stokes equations with a bigger rate of convergence than the plain Galerkin method. As a consequence, the difference between the Galerkin approximations and the solution of the linearized problem is an estimator of the Galerkin error. Moreover, since the Galerkin approximation of the evolutionary Navier–Stokes equations is also the Galerkin approximation of the linearized equations, any a posteriori estimator of the error in the Newton-type linearized Navier–Stokes equations is also an a posteriori error estimator of the full Navier–Stokes equations.