Summary: A new mixed variational formulation to the elliptic equation is given based on the less regularity of flux in practical problems, and the existence and uniqueness of solution to this formulation is shown in the framework of the saddle point problem by verifying the Ladyshenskaya-Babuška-Brezzi condition. Since in the new mixed variational formulation, the pressure belongs to the square integrable space instead of the classical \(\mathbf H\)(div), the choices of finite element pairs become simple and easy. Based on this new formulation, the corresponding conforming finite element approximation is addressed, and the existence and uniqueness of finite element solutions are obtained for \(P^2_0-P_1\) pairs consisting of piecewise constant elements for velocity and piecewise linear elements for pressure by proving the boundedness of the projection operator for velocity. Moreover, it is proven that the finite element approximation is optimal in some sense. Finally, some numerical experiments verify the effectiveness of and theoretical results for our method. This method can be extended to utilize the lowest equal-order finite element pair by adding some simple stabilization terms.