# The Mixed Finite Element Multigrid Preconditioned MINRES Method for Stokes Equations

Authors

1 University of Venda, Department of Mathematics,P Bag X5050, Thohoyandou 0950, South Africa

2 University of KwaZulu-Natal, Department of Mathematics, P Bag X01, Pietermaritzburg, Scottsville 3209, South Africa

Abstract

The study considers the saddle point problem arising from the mixed finite element discretization of the steady state Stokes equations. The saddle point problem is an indefinite system of linear equations, a feature that degrades the performance of any iterative solver. The heart of the study is the construction of fast, robust and effective iterative solution methods for such systems. Specific attention is given to the preconditioned MINRES solver PMINRES which is carefully treated for the solution of the Stokes equations. The study concentrates on the block preconditioner applied to the MINRES to effectively solve the whole coupled system. We combine iterative techniques with the MINRES as preconditioner approximations to produce an efficient solver for indefinite system of equations. We consider different preconditioner approximations of the building blocks of the preconditioner and compare their effects in accelerating the MINRES iterative scheme. We give a detailed overview of the algorithmic aspects and the theoretical convergence analysis of our solver. We study the MINRES method with the following preconditioner approximations: diagonal, multigrid v-cycle, preconditioned conjugate gradient and Chebyshev semi iteration methods. A comparative analysis of the preconditioner approximations show that the multigrid method is a suitable accelerator for the MINRES method. The application of the preconditioner becomes mandatory as evidenced by poor performance of the MINRES as compared to PMINRES. We study the problem in a two dimensional setting using the Hood-Taylor Q2 − Q1 stable pair of finite elements. The incompressible flow iterative solution software(IFISS) matlab toolbox is used to assemble the matrices. We present the numerical results to illustrate the efficiency and robustness of the MINRES scheme with the multigrid preconditioner.

Keywords