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

Abstract

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 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 H ¹- 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.

Keywords:

• Poisson equation
• mixed variational formulation
• stabilized conforming finite-element method
• two local Gauss integrations
• LBB condition

2000 AMS Subject Classifications :

• 65N12
• 65N30
• 46N20

Acknowledgements

This work was partially supported by NSFC with Nos 10971165, 10771167, 10926080, 10701101, 11071193.