Abstract
We present a new stabilized finite element method for incompressible flows based on Brezzi-Pitkäranta stabilized method. The stability and error estimates of finite element solutions are derived for classical one-level method. Combining the techniques of two-level discretizations, we propose two-level Stokes/Oseen/Newton iteration methods corresponding to three different linearization methods and show the stability and error estimates of these three methods. We also propose a new Newton correction scheme based on the above two-level iteration methods. Finally, some numerical experiments are given to support the theoretical results and to check the efficiency of these two-level iteration methods.
Highlights
In this paper, we consider steady Navier-Stokes equations with homogeneous Dirichlet boundary conditions:−μΔu + (u ⋅ ∇) u − ∇p = f, in Ω, div u = 0, in Ω, (1)u = 0, on ∂Ω, where Ω ⊂ R2 is a bounded convex domain with boundary ∂Ω. μ > 0 represents the viscous coefficient. u = (u1(x), u2(x)) denotes the velocity vector, p = p(x) the pressure, and f = (f1(x), f2(x)) the prescribed body force vector
Based on the Brezzi-Pitkaranta stabilized finite element method, in this paper, we solve the nonlinear Navier-Stokes equations on the coarse mesh with mesh size H in Step I and solve a linear system according to Stokes/Oseen/Newton iterative method on the fine mesh with mesh size h in Step II
It is obvious that if we choose H = O(h1/2), two-level method discussed in this paper provides the same convergence order as the classical one-level method
Summary
We consider steady Navier-Stokes equations with homogeneous Dirichlet boundary conditions:. We combine the Brezzi-Pitkaranta stabilized method, which is unconditionally stable [9], with techniques of two-level discretizations to solve the numerical solution of the problem (1) under the uniqueness condition. Based on the Brezzi-Pitkaranta stabilized finite element method, in this paper, we solve the nonlinear Navier-Stokes equations on the coarse mesh with mesh size H in Step I and solve a linear system according to Stokes/Oseen/Newton iterative method on the fine mesh with mesh size h in Step II. The finite element approximation solution (u⋆h , p⋆h) is solved in terms of Newton iterative scheme on the fine mesh in Step III. The numerical experiments are displaced to support the theoretical results
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