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