A mixed discontinuous Galerkin method for an unsteady incompressible Darcy equation

Abstract

We study a mixed discontinuous Galerkin (MDG) method for solving a time-dependent Darcy problem, which simulates an incompressible fluid such as water flowing in a rigid porous medium. The discretization of the unsteady Darcy problem relies on a backward Euler scheme for temporal variable and MDG method for spatial variables. Spatially semi-discrete and fully discrete schemes are analyzed. Existence and uniqueness of the numerical solutions are proved, and optimal order error estimates are derived for both the velocity and pressure variables. Finally, some test problems are provided to display the performance of the MDG method, and numerical results are reported to support the theoretical predictions.

Keywords:

• Mixed discontinuous Galerkin method
• unsteady Darcy flow
• incompressibility
• error estimates
• fully discrete scheme

AMS Classifications:

• 65N30
• 49J40

Disclosure statement

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

Additional information

Funding

This work was partially supported by the National Natural Science Foundation of China (Grant No. 11771350,11971377) and the Natural Science Foundation of Shaanxi Province (No. 2019JM-367).