A two-grid algorithm based on Newton iteration for the stream function form of the Navier-Stokes equations

In this paper, we propose a two-grid algorithm for solving the stream function formulation of the stationary Navier-Stokes equations. The algorithm is constructed by reducing the original system to one small, nonlinear system on the coarse mesh space and two similar linear systems (with same stiffness matrix but different right-hand side) on the fine mesh space. The convergence analysis and error estimation of the algorithm are given for the case of conforming elements. Furthermore, the algorithm produces a numerical solution with the optimal asymptotic H2-error. Finally, we give a numerical illustration to demonstrate the effectiveness of the two-grid algorithm for solving the Navier-Stokes equations.

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

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

[3]  Michel Fortin,et al.  Mixed and Hybrid Finite Element Methods , 2011, Springer Series in Computational Mathematics.

[4]  Howard C. Elman,et al.  Finite Elements and Fast Iterative Solvers: with Applications in Incompressible Fluid Dynamics , 2014 .

[5]  R. A. Nicolaides,et al.  Analysis of nonconforming stream function and pressure finite element spaces for the Navier-Stokes equations , 1989 .

[6]  William Layton,et al.  A multilevel mesh independence principle for the Navier-Stokes equations , 1996 .

[7]  R. Nicolaides,et al.  Finite element technique for optimal pressure recovery from stream function formulation of viscous flows , 1986 .

[8]  Faisal Fairag.  Numerical Computations of Viscous, Incompressible Flow Problems Using a Two-Level Finite Element Method , 2003, SIAM J. Sci. Comput..

[9]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[10]  X. Ye.  Two grid discretizations with backtracking of the stream function form of the Navier-Stokes equations , 1999, Appl. Math. Comput..

[11]  V. Girault,et al.  Two-grid finite-element schemes for the steady Navier-Stokes problem in polyhedra. , 2001 .

[12]  William Layton,et al.  Two-level discretizations of the stream function form of the navier-stokes equations , 1999 .

[13]  W. Hackbusch,et al.  Analysis of a damped nonlinear multilevel method , 1989 .

[14]  P. Raviart,et al.  Finite Element Approximation of the Navier-Stokes Equations , 1979 .

[15]  Max Gunzburger,et al.  Finite Element Methods for Viscous Incompressible Flows: A Guide to Theory, Practice, and Algorithms , 1989 .

[16]  Xiaoliang Cheng,et al.  A two-grid method based on Newton iteration for the Navier-Stokes equations , 2008 .

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

[18]  P. Hood,et al.  A numerical solution of the Navier-Stokes equations using the finite element technique , 1973 .

[19]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[20]  D. Rose,et al.  Analysis of a multilevel iterative method for nonlinear finite element equations , 1982 .

[21]  Lutz Tobiska,et al.  A Two-Level Method with Backtracking for the Navier--Stokes Equations , 1998 .

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

[23]  King Fahd,et al.  A TWO-LEVEL DISCRETIZATION METHOD FOR THE STREAMFUNCTION FORM OF THE NAVIER-STOKES EQUATIONS , 1998 .

[24]  Faisal Fairag.  A Two-Level Finite-Element Discretization of the Stream Function Form of the Navier-Stokes Equations , 1998 .

[25]  William Layton,et al.  A two-level discretization method for the Navier-Stokes equations , 1993 .