Chebyshev pseudospectral solution of the Stokes equations using finite element preconditioning

Abstract

The Stokes equations are solved by a Chebyshev pseudospectral method on a rectangular domain. As the resulting system of algebraic equations is very difficult to factorize, a preconditioning is designed using a finite element technique. The FEM solver constitutes the masterpiece of a Richardson iteration process. Several finite elements are investigated: the 9-nodes Lagrangian element /b Q/2-/b Q/1, the /b Q/1-/b Q/0 element, and the /b Q/1-/b Q/1 element due to Brezzi and Pitkaranta. An eigenvalue analysis is carried out in order to pinpoint the characteristic features of each precondition. It is shown that the /b Q/2-/b Q/1 element yields the best convergence results. The power of this choice is demonstrated on theoretical solutions and on the regularized square cavity problem.Anglai