A multigrid solver for a stabilized finite element discretization of the Stokes problem

1986 ◽  
pp. 1-6 ◽  
Author(s):  
E. M. Abdalass ◽  
J. F. Maitre ◽  
F. Musy
Keyword(s):  
Finite Element ◽  
Stokes Problem ◽  
Finite Element Discretization ◽  
Multigrid Solver ◽  
Element Discretization ◽  
Stabilized Finite Element

A TRUNCATED FOURIER/FINITE ELEMENT DISCRETIZATION OF THE STOKES EQUATIONS IN AN AXISYMMETRIC DOMAIN

2006 ◽  
Vol 16 (02) ◽  
pp. 233-263 ◽  
Author(s):  
Z. BELHACHMI ◽  
C. BERNARDI ◽  
S. DEPARIS ◽  
F. HECHT
Keyword(s):  
Finite Element ◽  
Stokes Equations ◽  
Stokes Problem ◽  
A Priori ◽  
Three Dimensional ◽  
Angular Variable ◽  
A Posteriori ◽  
Finite Element Discretization ◽  
Element Discretization ◽  
Good Agreement

We consider the Stokes problem in a three-dimensional axisymmetric domain and, by writing the Fourier expansion of its solution with respect to the angular variable, we observe that each Fourier coefficient satisfies a system of equations on the meridian domain. We propose a discretization of this problem which combines Fourier truncation and finite element methods applied to each of the two-dimensional systems. We give the detailed a priori and a posteriori analyses of the discretization and present some numerical experiments which are in good agreement with the analysis.

scholarly journals The Mixed Finite Element Multigrid Method for Stokes Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-12
Author(s):  
K. Muzhinji ◽  
S. Shateyi ◽  
S. S. Motsa
Keyword(s):  
Finite Element ◽  
Multigrid Method ◽  
Stokes Equations ◽  
Stokes Problem ◽  
Mixed Finite Element ◽  
Iterative Solvers ◽  
Algebraic Equations ◽  
Multigrid Solver ◽  
Element Discretization ◽  
Linear Algebraic Equations

The stable finite element discretization of the Stokes problem produces a symmetric indefinite system of linear algebraic equations. A variety of iterative solvers have been proposed for such systems in an attempt to construct efficient, fast, and robust solution techniques. This paper investigates one of such iterative solvers, the geometric multigrid solver, to find the approximate solution of the indefinite systems. The main ingredient of the multigrid method is the choice of an appropriate smoothing strategy. This study considers the application of different smoothers and compares their effects in the overall performance of the multigrid solver. We study the multigrid method with the following smoothers: distributed Gauss Seidel, inexact Uzawa, preconditioned MINRES, and Braess-Sarazin type smoothers. A comparative study of the smoothers shows that the Braess-Sarazin smoothers enhance good performance of the multigrid method. We study the problem in a two-dimensional domain using stable Hood-TaylorQ2-Q1pair of finite rectangular elements. We also give the main theoretical convergence results. We present the numerical results to demonstrate the efficiency and robustness of the multigrid method and confirm the theoretical results.

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