On error estimates of the fully discrete penalty method for the viscoelastic flow problem

Abstract

In this paper, a fully discrete finite element penalty method is presented for the two-dimensional viscoelastic flow problem arising in the Oldroyd model, in which the spatial discretization is based on the finite element approximation and the time discretization is based on the backward Euler scheme. Moreover, we provide the optimal error estimate for the numerical solution under some realistic assumptions. Finally, some numerical experiments are shown to illustrate the efficiency of the penalty method.

Keywords:

• viscoelastic flow
• Oldroyd model
• penalty method
• fully discrete scheme
• optimal error estimate

2000 AMS Subject Classifications :

• 35L70
• 65M30
• 76D05
• 78A10

Acknowledgements

This work was supported by the NSF of China (No. 10971166, No. 10901131), and the National High Technology Research and Development Program of China (863 Program, No. 2009AA01A135), the China Postdoctoral Science Foundation (No. 20070421155, No. 200801448), and the Natural Science Foundation of Xinjiang Province (No. 2010211B04). The authors thank the editor and reviewers for their criticism, valuable comments, and suggestions which helped to improve the results of this paper.