In this thesis we study BDM mixed finite element space and show that $L^2$ -error estimates of BDM mixed method are obtained for the approximation of pressure and the associated velocity.In order to produce a better approximation for the pressure, we discuss post-process based on the use of Lagrange multipliers to enforce interelement continuity. Furthermore, we establish and equivalence between given mixed methods and projection finite element method developed by Chen. We present implementation of first order BDM on rectangular grids.
