A stabilized multiple time step method for coupled Stokes-Darcy flows and transport model

Jingyuan Zhang; Ruikun Zhang; Xue Lin; Jingyuan Zhang; Ruikun Zhang; Xue Lin

doi:10.3934/math.20231091

AIMS Mathematics

2023, Volume 8, Issue 9: 21406-21438. doi: 10.3934/math.20231091

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A stabilized multiple time step method for coupled Stokes-Darcy flows and transport model

School of Mathematics and Physics, Qingdao University of Science and Technology, Qingdao, Shandong, 266061, China

Received: 24 May 2023 Revised: 19 June 2023 Accepted: 28 June 2023 Published: 05 July 2023
MSC : 65M12, 65M15, 65M60

A stabilized finite element algorithm with different time steps on different physical variables for the coupled Stokes-Darcy flows system with the solution transport is studied. The viscosity in the model is assumed to depend on the concentration. The nonconforming piecewise linear Crouzeix-Raviart element and piecewise constant are used to approximate velocity and pressure in the coupled Stokes-Darcy flows system, and conforming piecewise linear finite element is used to approximate concentration in the transport system. The time derivatives are discretized with different step sizes for the partial differential equations in these two systems. The existence and uniqueness of the approximate solution are unconditionally satisfied. A priori error estimates are established, which also provides a guidance on the ratio of time step sizes with respect to the ratio of the physical parameters. Numerical examples are presented to verify the theoretical results.
- coupled Stokes and Darcy flows,
- soulte transport,
- Crouzeix-Raviart finite element,
- multiple time step method,
- error estimates
Citation: Jingyuan Zhang, Ruikun Zhang, Xue Lin. A stabilized multiple time step method for coupled Stokes-Darcy flows and transport model[J]. AIMS Mathematics, 2023, 8(9): 21406-21438. doi: 10.3934/math.20231091

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Abstract

A stabilized finite element algorithm with different time steps on different physical variables for the coupled Stokes-Darcy flows system with the solution transport is studied. The viscosity in the model is assumed to depend on the concentration. The nonconforming piecewise linear Crouzeix-Raviart element and piecewise constant are used to approximate velocity and pressure in the coupled Stokes-Darcy flows system, and conforming piecewise linear finite element is used to approximate concentration in the transport system. The time derivatives are discretized with different step sizes for the partial differential equations in these two systems. The existence and uniqueness of the approximate solution are unconditionally satisfied. A priori error estimates are established, which also provides a guidance on the ratio of time step sizes with respect to the ratio of the physical parameters. Numerical examples are presented to verify the theoretical results.

References

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