Higher order isoparametric finite element solution of Stokes flow

A finite element method of the penalization type, using higher order shape functions with rectangular elements, is presented for the solution of the Stokes flow. The fluid is viscous and the flow is slow, steady and incompressible. The solution has been obtained in terms of velocity field while the pressure terms are eliminated using the penalty function approach. The degree of the finite element approximation can be increased as much as one needs with this higher order shape function formulation. The method is tested in a lid-driven square cavity and it is found that there is a relationship between the viscosity and penalty parameter. From one solution one can obtain all the other solutions only by arranging one of the parameters since the product of these parameters has been shown that is a fixed constant. Selected graphs of the velocity field are given showing these behaviors.