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AIMS Mathematics

2024, Volume 9, Issue 1: 453-480. doi: 10.3934/math.2024025
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Research article

Convergence analysis of Euler and BDF2 grad-div stabilization methods for the time-dependent penetrative convection model

  • College of Mathematics and Physics, Wenzhou University, Wenzhou 325035, China
  • Received: 31 October 2023 Revised: 20 November 2023 Accepted: 20 November 2023 Published: 30 November 2023

  • MSC : 36Q30, 65M60, 76M10

  • Based on the grad-div stabilization method, the first-order backward Euler and second-order BDF2 finite element schemes were studied for the approximations of the time-dependent penetrative convection equations. The proposed schemes are both unconditionally stable. We proved the error bounds of the velocity and temperature in which the constants are independent of inverse powers of the viscosity and thermal conductivity coefficients when the Taylor-Hood element and $ P_2 $ element are used in finite element discretizations. Finally, numerical experiments with high Reynolds numbers were shown to confirm the theoretical results.

    Citation: Weiwen Wan, Rong An. Convergence analysis of Euler and BDF2 grad-div stabilization methods for the time-dependent penetrative convection model[J]. AIMS Mathematics, 2024, 9(1): 453-480. doi: 10.3934/math.2024025

    Related Papers:

  • Based on the grad-div stabilization method, the first-order backward Euler and second-order BDF2 finite element schemes were studied for the approximations of the time-dependent penetrative convection equations. The proposed schemes are both unconditionally stable. We proved the error bounds of the velocity and temperature in which the constants are independent of inverse powers of the viscosity and thermal conductivity coefficients when the Taylor-Hood element and $ P_2 $ element are used in finite element discretizations. Finally, numerical experiments with high Reynolds numbers were shown to confirm the theoretical results.


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