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 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

Conforming Stokes Elements Yielding Divergence-Free Approximations on Quadrilateral Meshes

In this dissertation, we propose conforming finite element methods that yield divergence-free velocity approximations for the steady Stokes problem on cubical and quadrilateral meshes. In the first part, we construct the finite element spaces for the two-dimensional problem on rectangular grids. Then in the second part, we extend these spaces to n-dimensional spaces. We use discrete differential forms and smooth de Rham complexes to verify the stability and the conformity of the proposed methods, and the solenoidality of the velocity approximations. In the third part, we shift our focus from a dimensionwise extension to a meshwise improvement by introducing macro elements on general shape-regular quadrilateral meshes. By utilizing a smooth de Rham complex, we prove that the macro finite element method yields divergence-free velocity solutions, and with the construction of a Fortin operator, we validate the stability of the method. To improve the pressure approximation properties, we compute a post-processed pressure solution locally. In addition, we describe the implementation process of the (velocity) macro finite element. We show that the methods developed in this dissertation yield optimal convergence rates and present numerical experiments which are supportive of the theoretical results. Moreover, we provide experimental results of our method for the Navier-Stokes equations and show that the convergence rates are preserved