Efficient Multigrid Solvers for the Stokes Equations using Finite Elements

Mixed nite element application to solve the Stokes equations using di erent smoothers by mean of multigrid technique has been investigated. Multigrid technique's e ciency for positive de nite linear systems is proved but it is much more challenging in the case of saddle-point problems. The e ciency of a multigrid method is highly dependent on the smoother and the coarse grid solver. A two-dimensional square domain with di erent mixed nite element pairs is the framework for this work. The e ciency of two di erent smoothers, Vanka and Distributive Gauss-Seidel were investigated with utilizing P2/P1 and P2-iso-P1 nite element pairs, moreover the Vanka smoother was applied two a stabilized P1/P1 element discretization. The Pressure Correction method as a well-understood approach was the tool to validate the results of Vanka and Distributed Gauss-Seidel smoothers. Moreover an investigation on τ -extrapolation technique to raise the accuracy of linear nite elements to achieve the accuracy of quadratic nite elements was done. To this end τ -extrapolation technique was applied to a P1/P1 and P2-iso-P1 nite element pairs in order to obtain a better accuracy.