A second-order modular grad-div stabilized scheme for the Darcy–Brinkman model

Abstract

In this article, a fully discrete modular grad-div stabilized finite element scheme for the Darcy–Brinkman equations are considered. This fully discrete scheme includes two steps: the first step is a combination of a mixed finite element approximation for space discretization, the second-order backward differentiation formula for temporal discretization, and extrapolated treatments in linearization for the nonlinear terms. Then, in the second step, a modular grad-div stabilized technique is applied, which can improve solution accuracy without increasing computational time for large stabilized parameters. Moreover, we show that the scheme is unconditionally stable and convergent with second-order accuracy with respect to time step. Finally, several numerical examples are provided to support the derived theoretical results, and demonstrate the efficiency of the presented scheme.

KEYWORD:

• Darcy–Brinkman problem
• grad-div stabilized technique
• mixed finite element method
• second-order scheme

Acknowledgment

The authors would like to thank the editor and anonymous reviewers for their helpful comments and suggestions which lead to a considerably improved presentation.

Additional information

Funding

This work is sponsored by Natural Science Foundation of Xinjiang Uygur Autonomous Region (grant number 2021D01E11).