A posteriori error analysis of a mixed virtual element method for a nonlinear Brinkman model of porous media flow

Some aspects on the computational implementation of diverse terms arising in mixed virtual element formulations

“…In this section we present a numerical experiment in order to illustrate the performance of the mixed virtual element scheme (5.6) employing the Newton's iteration introduced in (5.8). It allows us to validate the operators introduced in Section 4, together with the numerical experiments presented in recent papers about mixed-VEM schemes, which utilized our implementation approach (see [15,17,16,26,25,32]). We begin by recalling from (5.10) that N stands for the total number of degrees of freedom (unknowns) of (5.8).…”

“…We end this paper by remarking some possible future directions. It would be interesting to explain some computational aspects about the adaptivity and the a posteriori error estimates for mixed virtual element schemes (see, e.g., [20,32]). In addition, in this work we used direct solvers for solving each lineal system.…”

“…Additionally, the analysis presented in [27] allows to study problems of the same nature as Navier-Stokes, such as the Boussinesq problem, where a mixed method of virtual elements was introduced and analyzed in [26]. For several other contributions on VEM and mixed-VEM we refer for instance to [2], [9], [12], [24], and [32].…”

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 posteriori virtual element method for the acoustic vibration problem

“…Recent research papers report interesting advantages of the VEM in the a posteriori error analysis and adaptivity for source problems. We refer to [7,16,17,37] and the references therein, for instance, for a further discussion. On the other hand, a posteriori error analysis for eigenproblems by VEM have been recently introduced in [40,33,35], where primal formulations in H 1 have been considered.…”