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

摘要

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