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.

Highlights

  • There are numerous works devoted to the development of efficient stable mixed finiteelement methods for solving the Navier-Stokes equations

  • The method we study in this paper is to combine the new stabilized finite-element method in 14 with the two-level method based on the Oseen iterative technique by the lowest-order conforming finite-element pairs Q1 − P0 and P1 − P0 for solving the 2D/3D stationary Navier-Stokes problems

  • We concentrate on the performance of the one-level finite-element method and Oseen two-level finite element method described in this paper

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Summary

Introduction

There are numerous works devoted to the development of efficient stable mixed finiteelement methods for solving the Navier-Stokes equations. The idea of the stabilized finite-element method based on the local pressure projection is derived from 14 for the Stokes equations This method differs from existing stabilization techniques and avoids approximation of derivatives, specification of mesh-dependent parameters, interface boundary data structures, and evaluated locally at the element level. This stabilized technique has been extended to solve the two-dimensional Navier-Stokes equation by Wang et al 15. The method we study in this paper is to combine the new stabilized finite-element method in 14 with the two-level method based on the Oseen iterative technique by the lowest-order conforming finite-element pairs Q1 − P0 and P1 − P0 for solving the 2D/3D stationary Navier-Stokes problems.

Functional Setting of the Navier-Stokes Equations
The Stabilized Finite-Element Method
Oseen Two-Level Stabilized Finite-Element Method
Numerical Examples
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