A Finite Element Method for the Stokes Problem on Quadrilateral Grids Yielding Divergence Free Approximations

In this thesis project, a pair of conforming, stable and divergence free finite \ud \ud elements for the Stokes problem on two dimensional rectangular grids with no-\ud \ud slip boundary conditions is constructed. Pointwise continuous Q3,2 x Q2,3 \ud \ud polynomials that are partially C1 at the vertices and Q2,2 polynomials that \ud \ud are continuous at the vertices are used as the functions forming the velocity and \ud \ud pressure spaces, respectively. In the construction of these finite element spaces, \ud \ud a Stokes complex is formed to verify the incompressibility of the velocity \ud \ud approximation. \ud \ud With the definition of appropriate norms and the use of the Piola transform, the \ud \ud inf-sup stability condition is satisfied on each rectangular element and then in \ud \ud the entire domain. Furthermore, by applying Nitsche's method to the problem \ud \ud and with the verification of the coercivity and continuity of the bilinear form, the \ud \ud existence and the uniqueness of the solution to the Stokes problem is justified. \u