Optimal order error estimates of a mixed discontinuous Galerkin method with BDF-2 scheme for transient Darcy flow

Abstract

In this paper, we study the numerical solution of Darcy flow via a mixed discontinuous Galerkin method. The two-step Euler backward formula is used for the time variable discretization and a mixed discontinuous Galerkin method is applied to spatial variable discretization. The stability results are established for our mixed discontinuous Galerkin numerical scheme in a fully discrete framework. Optimal error estimates for our unconditional scheme are derived for both the velocity and pressure variables. Finally, a numerical experiment is provided to approve the accuracy and efficiency of the proposed numerical method.

Keywords:

• Mixed discontinuous Galerkin method
• transient Darcy flow
• spatially semi-discrete scheme
• fully discrete scheme
• two-step backward differentiation formula (BDF-2)
• error estimate

Disclosure statement

The author(s) certify that there is no conflict of interest with any financial/research/academic organization, with regards to the content/research work discussed in the manuscript.

Additional information

Funding

This work was partially supported by the National Natural Science Foundation of China [grant number 1226106712361088], Inner Mongolia Natural Science Foundation of China [grant number 2021LHMS01006] and supported by the Fundamental Research Funds for Inner Mongolia University of Science & Technology.