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1. Highly efficient ensemble algorithms for computing the Stokes–Darcy equations

   https://doi.org/10.1016/j.cma.2023.116562  

   Jiang, Nan; Yang, Huanhuan (January 2024, Computer Methods in Applied Mechanics and Engineering)
2. Stochastic multiscale flux basis for Stokes-Darcy flows

   https://doi.org/10.1016/j.jcp.2019.109011  

   Ambartsumyan, Ilona; Khattatov, Eldar; Wang, ChangQing; Yotov, Ivan (January 2020, Journal of Computational Physics)
3. Analysis of a diffuse interface method for the Stokes-Darcy coupled problem

   https://doi.org/10.1051/m2an/2023062  

   Bukač, Martina; Muha, Boris; Salgado, Abner J (September 2023, ESAIM: Mathematical Modelling and Numerical Analysis)

   We consider the interaction between a free flowing fluid and a porous medium flow, where the free flowing fluid is described using the time dependent Stokes equations, and the porous medium flow is described using Darcy’s law in the primal formulation. To solve this problem numerically, we use a diffuse interface approach, where the weak form of the coupled problem is written on an extended domain which contains both Stokes and Darcy regions. This is achieved using a phase-field function which equals one in the Stokes region and zero in the Darcy region, and smoothly transitions between these two values on a diffuse region of width (ϵ) around the Stokes-Darcy interface. We prove convergence of the diffuse interface formulation to the standard, sharp interface formulation, and derive rates of convergence. This is performed by deriving a priori error estimates for discretizations of the diffuse interface method, and by analyzing the modeling error of the diffuse interface approach at the continuous level. The convergence rates are also shown computationally in a numerical example. 

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4. On Stokes--Ritz Projection and Multistep Backward Differentiation Schemes in Decoupling the Stokes--Darcy Model

   https://doi.org/10.1137/16M1099601  

   Gunzburger, Max; He, Xiaoming; Li, Buyang (January 2018, SIAM Journal on Numerical Analysis)

   null (Ed.)
5. Nonconforming time discretization based on Robin transmission conditions for the Stokes–Darcy system

   https://doi.org/10.1016/j.amc.2021.126602  

   Hoang, Thi-Thao-Phuong; Kunwar, Hemanta; Lee, Hyesuk (January 2022, Applied mathematics and computation)

   We consider a space-time domain decomposition method based on Schwarz waveform relaxation (SWR) for the time-dependent Stokes-Darcy system. The coupled system is formulated as a time-dependent interface problem based on Robin-Robin transmission conditions, for which the decoupling SWR algorithm is proposed and proved for the convergence. In this approach, the Stokes and Darcy problems are solved independently and globally in time, thus allowing the use of different time steps for the local problems. Numerical tests are presented for both non-physical and physical problems with various mesh sizes and time step sizes to illustrate the accuracy and efficiency of the proposed method. 

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