An Oseen Two‐Level Stabilized Mixed Finite‐Element Method for the 2D/3D Stationary Navier‐Stokes Equations

Abstract

We investigate an Oseen two‐level stabilized finite‐element method based on the local pressure projection for the 2D/3D steady Navier‐Stokes equations by the lowest order conforming finite‐element pairs (i.e., Q1 − P0 and P1 − P0). Firstly, in contrast to other stabilized methods, they are parameter free, no calculation of higher‐order derivatives and edge‐based data structures, implemented at the element level with minimal cost. In addition, the Oseen two‐level stabilized method involves solving one small nonlinear Navier‐Stokes problem on the coarse mesh with mesh size H, a large general Stokes equation on the fine mesh with mesh size h = O(H)2. The Oseen two‐level stabilized finite‐element method provides an approximate solution (uh, ph) with the convergence rate of the same order as the usual stabilized finite‐element solutions, which involves solving a large Navier‐Stokes problem on a fine mesh with mesh size h. Therefore, the method presented in this paper can save a large amount of computational time. Finally, numerical tests confirm the theoretical results. Conclusion can be drawn that the Oseen two‐level stabilized finite‐element method is simple and efficient for solving the 2D/3D steady Navier‐Stokes equations.