A Fully Discrete Stabilized Mixed Finite Element Method for Parabolic Problems

Abstract

In this article, we present a new stabilized mixed finite element method based on the less regularity of flux (velocity) for the two-dimensional parabolic problems in practice. The method combines the Crank-Nicolson scheme with a stabilized mixed finite element method which is based on the velocity projection stabilization method by using the lowest equal-order pair for the velocity and pressure. It is shown that the proposed fully discrete stabilized finite element method results in the optimal error estimates in L ²- and H ¹-norm for the pressure and suboptimal error estimate in L ²-norm for the velocity. Finally, we give some numerical experiments to verify the efficiency and theoretical results of this method.

Acknowledgments

This work is in part supported by the NSF of China (nos. 61163027, 11126112, and 11271313), the China Postdoctoral Science Foundation (nos. 201104702 and 2012M512056), the Key Project of Chinese Ministry of Education (no. 212197), the Project of Special Training of the Minority Nationality in Xinjiang (no. 201123117), and the Doctoral Foundation of Xinjiang University (no. BS110101).

The authors would like to thank the editor and referees for their valuable comments and suggestions which helped us to improve the results of this article.