Two-level Stabilized Finite Element Methods for the Steady Navier–Stokes Problem

Abstract

In this article, the two-level stabilized finite element formulations of the two-dimensional steady Navier–Stokes problem are analyzed. A macroelement condition is introduced for constructing the local stabilized formulation of the steady Navier–Stokes problem. By satisfying this condition the stability of the Q1−P0 quadrilateral element and the P1−P0 triangular element are established. Moreover, the two-level stabilized finite element methods involve solving one small Navier–Stokes problem on a coarse mesh with mesh size H, a large Stokes problem for the simple two-level stabilized finite element method on a fine mesh with mesh size h=O(H2) or a large general Stokes problem for the Newton two-level stabilized finite element method on a fine mesh with mesh size h=O(|log h|1/2H3). The methods we study provide an approximate solution (uh,ph) with the convergence rate of same order as the usual stabilized finite element solution, which involves solving one large Navier–Stokes problem on a fine mesh with mesh size h. Hence, our methods can save a large amount of computational time.