Two-level stabilized finite element method for the transient Navier–Stokes equations

Abstract

In this article, a two-level stabilized finite element method based on two local Gauss integrations for the two-dimensional transient Navier–Stokes equations is analysed. This new stabilized method presents attractive features such as being parameter-free, or being defined for nonedge-based data structures. Some new a priori bounds for the stabilized finite element solution are derived. The two-level stabilized method involves solving one small Navier–Stokes problem on a coarse mesh with mesh size 0<H<1, and a large linear Stokes problem on a fine mesh with mesh size 0<h≪H. A H 1-optimal velocity approximation and a L 2-optimal pressure approximation are obtained. If we choose h=O(H 2), the two-level method gives the same order of approximation as the standard stabilized finite element method.