Newton Iterative Parallel Finite Element Algorithm for the Steady Navier-Stokes Equations

A combination method of the Newton iteration and parallel finite element algorithm is applied for solving the steady Navier-Stokes equations under the strong uniqueness condition. This algorithm is motivated by applying the Newton iterations of m times for a nonlinear problem on a coarse grid in domain Ω and computing a linear problem on a fine grid in some subdomains Ωj⊂Ω with j=1,…,M in a parallel environment. Then, the error estimation of the Newton iterative parallel finite element solution to the solution of the steady Navier-Stokes equations is analyzed for the large m and small H and h≪H. Finally, some numerical tests are made to demonstrate the the effectiveness of this algorithm.

[1]  Jinchao Xu.  Two-grid Discretization Techniques for Linear and Nonlinear PDEs , 1996 .

[2]  R. Temam.  Navier-Stokes Equations , 1977 .

[3]  Jinchao Xu,et al.  Local and Parallel Finite Element Algorithms Based on Two-Grid Discretizations for Nonlinear Problems , 2001, Adv. Comput. Math..

[4]  Yi-chen Ma,et al.  Local and parallel finite element algorithms based on two-grid discretization for the stream function form of Navier-Stokes equations , 2006, Appl. Math. Comput..

[5]  Jinchao Xu,et al.  Local and parallel finite element algorithms based on two-grid discretizations , 2000, Math. Comput..

[6]  Jinchao Xu,et al.  Local and Parallel Finite Element Algorithms for Eigenvalue Problems , 2002 .

[7]  Ping Wang,et al.  On the Monotonicity of (k;g,h)-graphs , 2002 .

[8]  Yinnian He,et al.  Convergence of three iterative methods based on the finite element discretization for the stationary Navier–Stokes equations☆ , 2009 .

[9]  Jinchao Xu,et al.  A Novel Two-Grid Method for Semilinear Elliptic Equations , 1994, SIAM J. Sci. Comput..

[10]  马飞遥,et al.  Local and parallel finite element algorithms based on two-grid discretization for steady Navier-Stokes equations , 2007 .

[11]  Jinchao Xu,et al.  Local and parallel finite element algorithms for the stokes problem , 2008, Numerische Mathematik.

[12]  Vivette Girault,et al.  Finite Element Methods for Navier-Stokes Equations - Theory and Algorithms , 1986, Springer Series in Computational Mathematics.