The mixed finite element multigrid method for stokes equations.

K Muzhinji,S Shateyi,S S Motsa

doi:10.1155/2015/460421

K Muzhinji, S Shateyi + Show 1 more

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https://doi.org/10.1155/2015/460421

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Abstract

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-Taylor Q 2-Q 1 pair 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.

Highlights

This study considers the numerical solution of the large scale linear algebraic system arising from the discretization of the partial differential equations
In this paper we consider multigrid methods for solving linear indefinite algebraic system of equations arising from the mixed finite element discretization of the steady state Stokes problem:
We have introduced four smoother iterative methods for both multigrids V-cycle and W-cycle to solve the indefinite systems emanating for the mixed finite element discretization of the Stokes problem

Summary

Introduction

This study considers the numerical solution of the large scale linear algebraic system arising from the discretization of the partial differential equations. The discretization is achieved by the finite element method. For positive definite linear systems, linked to the Poisson equations, the multigrid (MGM) methods are proven to be the most effective and fast methods [1, 2]. It is more challenging for linear indefinite algebraic systems. In this paper we consider multigrid methods for solving linear indefinite algebraic system of equations arising from the mixed finite element discretization of the steady state Stokes problem:. The problem is considered with (1)–(3) defined on the domain Ω ⊆ R2 with boundary ∂Ω

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