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<p>A grad-div stabilized difference finite element method is constructed for the 3D stationary Stokes–Darcy equations. This stabilized method consists of finite element discretization in (<i>x</i>,&#xa0;<i>y</i>)-plane and finite difference discretization in <i>z</i>-direction. It transmits finite element solution of the 3D Stokes–Darcy equations in the direction of (<i>x</i>,&#xa0;<i>y</i>,&#xa0;<i>z</i>) into a series of finite element solution of the 2D Stokes–Darcy equations in the direction of (<i>x</i>,&#xa0;<i>y</i>), and thereby decreases the complexity of the 3D discretization and allows code reuse. Moreover, numerical analyses are developed, which show that the proposed stabilized method is stable and convergent. Numerical studies support the theoretical results and find that adding a grad-div stabilization term to the difference finite element approximation has a stabilizing effect for small viscosity.</p>

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A difference finite element method for the 3D stationary Stokes–Darcy equations with grad-div stabilization

  • 期刊论文

  • 发表日期：

    2025年6月27日

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  • Qi Zhang,
  • Pengzhan Huang,
  • Yinnian He

摘要

A grad-div stabilized difference finite element method is constructed for the 3D stationary Stokes–Darcy equations. This stabilized method consists of finite element discretization in (x, y)-plane and finite difference discretization in z-direction. It transmits finite element solution of the 3D Stokes–Darcy equations in the direction of (x, y, z) into a series of finite element solution of the 2D Stokes–Darcy equations in the direction of (x, y), and thereby decreases the complexity of the 3D discretization and allows code reuse. Moreover, numerical analyses are developed, which show that the proposed stabilized method is stable and convergent. Numerical studies support the theoretical results and find that adding a grad-div stabilization term to the difference finite element approximation has a stabilizing effect for small viscosity.