This paper is concerned with non-uniform fully-mixed FEMs for dynamic coupled Stokes–Darcy model with the well-known Beavers–Joseph–Saffman (BJS) interface condition. In particular, a decoupled algorithm with the lowest-order mixed non-uniform FE approximations (MINI for the Stokes equation and RT0-DG0 for the Darcy equation) and the classical Nitsche-type penalty is studied. The method with the combined approximation of different orders is commonly used in practical simulations. However, the optimal error analysis of methods with non-uniform approximations for the coupled Stokes–Darcy flow model has remained challenging, although the analysis for uniform approximations has been well done. The key question is how the lower-order approximation to the Darcy flow influences the accuracy of the Stokes solution through the interface condition. In this paper, we prove that the decoupled algorithm provides a truly optimal convergence rate in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$L^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>-norm in spatial direction: <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$O(h^2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>h</mi> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for Stokes velocity and O(h) for Darcy flow in the coupled Stokes–Darcy model. This implies that the lower-order approximation to the Darcy flow does not pollute the accuracy of numerical velocity for Stokes flow. The analysis presented in this paper is based on a well-designed Stokes–Darcy Ritz projection and given for a dynamic coupled model. The optimal error estimate holds for more general combined approximations and more general coupled models, including the corresponding model of steady-state Stokes–Darcy flows and the model of coupled dynamic Stokes and steady-state Darcy flows. Numerical results confirm our theoretical analysis and show that the decoupled algorithm is efficient.

Optimal Analysis of Penalized Lowest-Order Mixed FEMs for the Stokes–Darcy Model

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发表日期：

2025年11月16日

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Luling Cao,
Weiwei Sun

摘要

This paper is concerned with non-uniform fully-mixed FEMs for dynamic coupled Stokes–Darcy model with the well-known Beavers–Joseph–Saffman (BJS) interface condition. In particular, a decoupled algorithm with the lowest-order mixed non-uniform FE approximations (MINI for the Stokes equation and RT0-DG0 for the Darcy equation) and the classical Nitsche-type penalty is studied. The method with the combined approximation of different orders is commonly used in practical simulations. However, the optimal error analysis of methods with non-uniform approximations for the coupled Stokes–Darcy flow model has remained challenging, although the analysis for uniform approximations has been well done. The key question is how the lower-order approximation to the Darcy flow influences the accuracy of the Stokes solution through the interface condition. In this paper, we prove that the decoupled algorithm provides a truly optimal convergence rate in $L^2$ $L^{2}$ -norm in spatial direction: $O(h^2)$ $O (h^{2})$ for Stokes velocity and O(h) for Darcy flow in the coupled Stokes–Darcy model. This implies that the lower-order approximation to the Darcy flow does not pollute the accuracy of numerical velocity for Stokes flow. The analysis presented in this paper is based on a well-designed Stokes–Darcy Ritz projection and given for a dynamic coupled model. The optimal error estimate holds for more general combined approximations and more general coupled models, including the corresponding model of steady-state Stokes–Darcy flows and the model of coupled dynamic Stokes and steady-state Darcy flows. Numerical results confirm our theoretical analysis and show that the decoupled algorithm is efficient.