A new stabilized mixed finite-element method for Poisson equation based on two local Gauss integrations for linear element pair

In this paper, we give a new mixed variational formulation to the Poisson equation based on the less regularity of flux(velocity) in practice, and show the existence and uniqueness of the solution to this saddle point problem. Based on this new formulation, we address its corresponding stabilization conforming the finite-element approximation for P₁² - P₁ finite-element pairs based on two local Gauss integrations for velocity, and give the finite-element solution's existence and uniqueness. Moreover, we obtain that the approximation of pressure p is optimal in H¹- and L ²-norms, the approximation of velocity u is suboptimal in H ¹-norm. Finally, we give some numerical experiment to verify the theoretical results.