In this paper, a two-grid decoupled penalty finite element method has been constructed for solving the mixed Stokes–Darcy model. We first introduce a penalty Stokes–Darcy model based on the penalty method at the continuous level and then show its solution converges to the original one as where the penalty parameter is . Then a two-grid method is used to decouple the penalty model. On the coarse mesh, it requires to solve the penalty model. On the fine mesh, we only need to solve two independent subproblems, while we avoid solving a saddle point problem in the fluid region due to the penalty method. We also provide the error estimates in which we prove the optimal convergence order with . In the final, we perform the numerical tests which indicate that our proposed scheme is effective and can achieve the same accuracy as the directly coupled finite element method.

中文翻译：

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求解平稳Stokes-Darcy问题的二网格解耦罚分有限元法


本文构建了一种两网格解耦罚分有限元方法来求解混合斯托克斯-达西模型。我们首先在连续水平上引入基于惩罚方法的惩罚Stokes-Darcy模型，然后证明其解收敛于原始解，其中惩罚参数为 。然后采用二网格方法对惩罚模型进行解耦。在粗网格上，需要求解惩罚模型。在细网格上，我们只需要解决两个独立的子问题，同时由于惩罚方法，我们避免了解决流体区域中的鞍点问题。我们还提供了误差估计，其中我们用 证明了最佳收敛顺序。最后，我们进行了数值试验，结果表明我们提出的方案是有效的，并且可以达到与直接耦合有限元法相同的精度。