A virtual element method for the coupled Stokes–Darcy problem with the Beaver–Joseph–Saffman interface condition

The virtual element method

“…Beirão da Veiga, Canuto, Nochetto and Vacca (2021) analysed a model fluid interaction problem using mesh cutting techniques in combination with the above VEM approach. There have also been other developments of the VEM for fluid mechanics problems outside the divergence-free framework, some examples being nonconforming methods (Cangiani, Gyrya and Manzini 2016, Liu, Li and Chen 2017, Zhao, Zhang, Mao and Chen 2020, Liu, Li and Chen 2019, non-standard mixed formulations (Cáceres, Gatica and Sequeira 2017, Gatica, Munar and Sequeira 2018b, Cáceres and Gatica 2016, Munar and Sequeira 2020, Gatica, Munar and Sequeira 2018a, Cáceres, Gatica and Sequeira 2018 and other derivations Wang 2019, Wang, Wang andHe 2020). Finally, a few references about the application of other polytopal technologies -such as polygonal FEMs, polygonal discontinuous Galerkin (DG), hybrid high-order (HHO) and hybridizable discontinuous Galerkin (HDG) -to fluid mechanic problems are those of Natarajan (2020), Botti, Di Pietro andDroniou (2018), Di Pietro andKrell (2018), Aghili and Di Pietro (2018), Castañón Quiroz and Di Pietro (2020), Lipnikov, Vassilev and Yotov (2014), Cockburn, Fu and Qiu (2017), Antonietti, Verani, Vergara and Zonca (2019) and Antonietti, Mascotto, Verani and Zonca (2022), while some references (among the many) on FEM divergence-free and pressure-robust methods are Guzmán and Scott (2019) and Neilan (2014, 2018), and Gauger, Linke and Schroeder (2019), Linke and Merdon (2016b), John et al (2017) and Linke and Merdon (2016a).…”

A virtual element discretization for the time dependent Navier–Stokes equations in stream-function formulation

“…Indeed, by avoiding the explicit construction of the local basis functions, the VEM can easily handle general polygons/polyhedrons without complex integrations on the element (see [7] for details on the coding aspects of the method). The VEM has been applied successfully for problems in fluid mechanics; see for instance [1,10,18,24,26,30,34,33,17,39,41], where Stokes, Brinkman, Stokes-Darcy and Navier-Stokes equations have been recently developed.…”

A Robin-type domain decomposition method for a novel mixed-type DG method for the coupled Stokes-Darcy problem

“…As it is well known, the standard mixed formulations for the Stokes equations and Darcy equations earn different compatibility conditions, thus a straightforward application of the existing solvers for the Stokes equations and Darcy equations may not be feasible. To this end, a great amount of effort has been devoted to developing accurate and efficient numerical schemes for the coupled Stokes-Darcy problem, and a non-exhaustive list of these approaches include Lagrange multiplier methods [21,17,32,18], weak Galerkin method [9,22], strongly conservative methods [20,16], stabilized mixed finite element method [28,24], discontinuous Galerkin (DG) methods [26,34], virtual element method [23,33], a lowest-order staggered DG method [37] and penalty methods [38]. The coupled Stokes-Darcy problem describes multiphysics phenomena, and involves a Stokes subproblem and a Darcy subproblem, it is thus natural to resort to domain decomposition methods, which allows one to solve the coupled system sequentially with a low computational cost.…”