A convergence analysis of semi-discrete and fully-discrete nonconforming FEM for the parabolic obstacle problem

Abstract

We propose and analyse a nonconforming finite element method for numerical approximation of the solution of a parabolic variational inequality associated with general obstacle. In this article, we carry out the error analysis for both the semi-discrete and fully-discrete schemes. We use the backward Euler method for time discretization and the lowest order Crouzeix-Raviart nonconforming finite element method for space discretization. The main motivation for the space discretization with Crouzeix-Raviart nonconforming finite element method to the parabolic obstacle problem is that it gives a natural $H^1$-stable interpolation which is commutative with the time derivative. By taking full advantage of this commutative property together with $H^1$-stability of interpolation, we derive an error estimate of the order $\mathcal{O}\left(h\right)$ for semi-discrete scheme, and error estimate of order $\mathcal{O}(h+{\left(\log \frac{1}{\mathrm{\Delta }t}\right)}^{1/4}{\mathrm{\Delta }t}^{3/4})$ for fully-discrete scheme in a certain norm defined precisely in the article.

Keywords:

• Finite element
• nonconforming
• parabolic obstacle problem
• variational inequalities
• backward Euler

2010 Mathematics Subject Classifications:

• 65N30
• 65N15

Acknowledgements

The author would like to thank Thirupathi Gudi for some useful comments and suggestions. Also the author would like to thank anonymous Referee and Editor for their helpful and constructive comments that lead to the improvement of this article.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by Council of Scientific and Industrial Research, India [RP03792G].