A divergence free weak virtual element method for the Stokes–Darcy problem on general meshes

Wang, Gang; Wang, Feng; Chen, Long; He, Ya‐Ling

doi:10.1016/j.cma.2018.10.022

Cited by 44 publications

(11 citation statements)

References 46 publications

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“…The virtual element method (VEM) is an increasingly popular tool in the approximation to solutions of fluido-static and dynamic problems based on polygonal/polyhedral meshes. In particular we recall: the very first paper on low-order VEM for Stokes [2]; its high-order conforming [11] and nonconforming versions [20,34]; conforming [12] and nonconforming VEM for the Navier-Stokes equation [33]; mixed VEM for the pseudo-stress-velocity formulation of the Stokes problem [17]; mixed VEM for quasi-Newtonian flows [19]; mixed VEM for the Navier-Stokes equation [24]; other variants of the VEM for the Darcy problem [18,45,47]; analysis of the Stokes complex in the VEM framework [9,13]; a stabilized VEM for the unsteady incompressible Navier-Stokes equations [30]; implementation details [23].…”

Section: Introductionmentioning

confidence: 99%

p- and hp- virtual elements for the Stokes problem

2021

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We analyse the p- and hp-versions of the virtual element method (VEM) for the Stokes problem on polygonal domains. The key tool in the analysis is the existence of a bijection between Poisson-like and Stokes-like VE spaces for the velocities. This allows us to re-interpret the standard VEM for Stokes as a VEM, where the test and trial discrete velocities are sought in Poisson-like VE spaces. The upside of this fact is that we inherit from Beirão da Veiga et al. (Numer. Math. 138(3), 581–613, 2018) an explicit analysis of best interpolation results in VE spaces, as well as stabilization estimates that are explicit in terms of the degree of accuracy p of the method. We prove exponential convergence of the hp-VEM for Stokes problems with regular right-hand sides. We corroborate the theoretical estimates with numerical tests for both the p- and hp-versions of the method.

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Section: Introductionmentioning

confidence: 99%

p- and hp- virtual elements for the Stokes problem

2021

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“…In practice, the stabilized term can be computed by the d.o.f. The VEM approach has been successfully applied to several classes of partial differential equations, such as the elliptic interface problem, 20 the Stokes and Navier–Stokes problems, 21,22 the Stokes–Darcy problem, 23 and many others.…”

Section: Introductionmentioning

confidence: 99%

Least‐squares virtual element method for the convection‐diffusion‐reaction problem

Wang

2021

Numerical Meth Engineering

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In this paper, we introduce a least‐squares virtual element method for the convection‐diffusion‐reaction problem in mixed form. We use the H(div) virtual element and continuous virtual element to approximate the flux and the primal variables, respectively. The method allows for the use of very general polygonal meshes. Optimal order a priori error estimates are established for the flux and the primal variables in suitable norms. The least‐squares method offers an efficient a posteriori error estimator without extra effort. Moreover, the hanging nodes are naturally treated in the virtual element method, which provides the high flexibility in mesh refinement because the local mesh postprocessing is never required. Both attractive features motivate us to develop the a posteriori error estimate of the method. Numerical experiments are shown to illustrate the accuracy of the theoretical analysis and demonstrate that the adaptive mesh refinement driven by the proposed estimator can efficiently capture the boundary and the interior layers.

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“…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.…”

Section: Introductionmentioning

confidence: 99%

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

Zhao¹

2021

Preprint

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In this paper, we first propose and analyze a novel mixed-type DG method for the coupled Stokes-Darcy problem on simplicial meshes. The proposed formulation is locally conservative. A mixed-type DG method in conjunction with the stress-velocity formulation is employed for the Stokes equations, where the symmetry of stress is strongly imposed. The staggered DG method is exploited to discretize the Darcy equations. As such, the discrete formulation can be easily adapted to account for the Beavers-Joseph-Saffman interface conditions without introducing additional variables. Importantly, the continuity of normal velocity is satisfied exactly at the discrete level. A rigorous convergence analysis is performed for all the variables. Then we devise and analyze a domain decomposition method via the use of Robin-type interface boundary conditions, which allows us to solve the Stokes subproblem and the Darcy subproblem sequentially with low computational costs. The convergence of the proposed iterative method is analyzed rigorously. In particular, the proposed iterative method also works for very small viscosity coefficient. Finally, several numerical experiments are carried out to demonstrate the capabilities and accuracy of the novel mixed-type scheme, and the convergence of the domain decomposition method.

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A divergence free weak virtual element method for the Stokes–Darcy problem on general meshes

Cited by 44 publications

References 46 publications

p- and hp- virtual elements for the Stokes problem

p- and hp- virtual elements for the Stokes problem

Least‐squares virtual element method for the convection‐diffusion‐reaction problem

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

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