A virtual element method for the miscible displacement of incompressible fluids in porous media

Veiga, L. Beirão da; Pichler, Alexander; Vacca, Giuseppe

doi:10.1016/j.cma.2020.113649

Cited by 24 publications

(11 citation statements)

References 63 publications

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“…Also, we are interested in pursuing some efforts for a deeper mathematical treatise of memory terms in the uptake function, recently investigated, for instance, in Wu et al (2020). Furthermore, we will investigate the approximation of non-steady solutions to (3) in the frame of Filippov theory, making use also of recent VEM approaches (da Veiga et al 2021). Finally, a challenge would be to integrate such physically based models into agricultural DSS as Zaza et al (2018), Friedman et al (2016).…”

Section: Discussionmentioning

confidence: 99%

Optimizing Water Consumption in Richards’ Equation Framework with Step-Wise Root Water Uptake: A Simplified Model

et al. 2022

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This work arises from the need of exploring new features for modeling and optimizing water consumption in irrigation processes. In particular, we focus on water flow model in unsaturated soils, accounting also for a root water uptake term, which is assumed to be discontinuos in the state variable. We investigate the possibility of accomplishing such optimization by computing the steady solutions of a $$\theta$$ θ -based Richards equation revised as equilibrium points of the ODEs system resulting from a numerical semi-dicretization in the space; after such semi-discretization, these equilibrium points are computed exactly as the solutions of a linear system of algebraic equations: the case in which the equilibrium lies on the threshold for the uptake term is of particular interest, since the system considerably simplifies. In this framework, the problem of minimizing the water waste below the root level is investigated. Numerical simulations are provided for representing the obtained results. Article Highlights Root water uptake is modelled in a Richards’ equation framework with a discontinuous sink term. After a proper semidiscretization in space, equilibrium points of the resulting nonlinear ODE system are computed exactly. The proposed approach simplifies a control problem for optimizing water consumption.

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Section: Discussionmentioning

confidence: 99%

Optimizing Water Consumption in Richards’ Equation Framework with Step-Wise Root Water Uptake: A Simplified Model

et al. 2022

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“…We remark that convergence analysis of VEM can be carried out similarly as in finite element methods by introducing certain projection operators onto the space of polynomial functions. Recently, VEM has been also developed in [12,14,24,25,26] for the Biot's equation and for nonlinear problems in [9,13].…”

Section: Introductionmentioning

confidence: 99%

Virtual Element Approximations for Two Species Model of the Advection-Diffusion-Reaction in Poroelastic Media

Verma

Kumar

2022

Mathematical Modelling and Analysis

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This paper proposes virtual element methods for approximating the mathematical model consisting of coupled poroelastic and Advection-Diffusion-Reaction (ADR) equations. The space discretization relies on virtual element spaces containing piecewise linear polynomials as well as non-polynomials for displacement, pressure and concentrations, and piecewise constant for total pressure; a backwardEuler scheme is employed for the approximation of time derivative. Using standard techniques of explicit schemes, the well-posedness of the resultant fully discrete scheme is derived. Moreover, under certain regularity assumptions on the mesh, optimal apriori error estimates are established by introducing suitable projection operators. Several numerical experiments are presented to validate the theoretical convergence rate and exhibit the proposed scheme’s performance.

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“…The possibility of using general polytopal meshes makes VEM suitable for diffusion problems, for instance by making much easier to adapt to complex geometry of the data (such as in basin and reservoir simulations) and to irregularities of the solution. The VEM literature on the diffusion-reaction-convection problem is indeed very wide, covering primal and mixed methods, conforming and non-conforming schemes, ranging from foundation/theoretical contributions to more applicative articles; a very short representative list being [9,8,20,5,24,10,16,14,30,31,12,26]. Some examples of other numerical methods for the diffusion-reactionconvection problem that can handle polytopal meshes are [27,28,4,22,3,21].…”

Section: Introductionmentioning

confidence: 99%

SUPG-stabilized virtual elements for diffusion-convection problems: a robustness analysis

et al. 2021

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The objective of this contribution is to develop a convergence analysis for SUPG-stabilized Virtual Element Methods in diffusion-convection problems that is robust also in the convection dominated regime. For the original method introduced in [Benedetto et al, CMAME 2016] we are able to show an “almost uniform” error bound (in the sense that the unique term that depends in an unfavourable way on the parameters is damped by a higher order mesh-size multiplicative factor). We also introduce a novel discretization of the convection term that allows us to develop error estimates that are fully robust in the convection dominated cases. We finally present some numerical result.

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A virtual element method for the miscible displacement of incompressible fluids in porous media

Cited by 24 publications

References 63 publications

Optimizing Water Consumption in Richards’ Equation Framework with Step-Wise Root Water Uptake: A Simplified Model

Optimizing Water Consumption in Richards’ Equation Framework with Step-Wise Root Water Uptake: A Simplified Model

Virtual Element Approximations for Two Species Model of the Advection-Diffusion-Reaction in Poroelastic Media

SUPG-stabilized virtual elements for diffusion-convection problems: a robustness analysis

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