Gradient Schemes: A Generic Framework for the Discretisation of Linear, Nonlinear and Nonlocal Elliptic and Parabolic Equations

The gradient scheme family, which includes the conforming and mixed finite elements as well as the mimetic mixed hybrid family, is used for the approximation of Richards equation and the two-phase flow problem in heterogeneous porous media. We prove the convergence of the approximate saturation and of the approximate pressures and approximate pressure gradients thanks to monotony and compactness arguments under an assumption of non-degeneracy of the phase relative permeabilities. Strong convergence results stem from the convergence of the norms of the gradients of pressures, which demand handling the nonlinear time term. Numerical results show the efficiency on these problems of a particular gradient scheme, called the Vertex Approximate Gradient scheme.

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“…Remark 3.1 (The scheme for Richards' equation) The scheme is obtained by Scheme (14), replacing (14d) by…”

Section: Remark 212 An Important Example Of Space-time Discretisationmentioning

confidence: 99%

“…x ∈ Ω j , and all q ∈ R and j ∈ J. We also have (14), and there exists C 1 > 0, only depending on the data introduced in Hypotheses (4), on any C P ∈ (0, +∞) greater than C D (defined by (8)), and on…”

Section: Remark 212 An Important Example Of Space-time Discretisationmentioning

confidence: 99%

Gradient schemes for two‐phase flow in heterogeneous porous media and Richards equation

Eymard¹,

Guichard

Herbin

et al. 2013

Z Angew Math Mech

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“…We follow here the same approach used in [3]. Define the mapping T : v −→ u where for any v ∈ X D,0 , u ∈ K D is defined as the solution to…”

Section: Theorem 2 (Convergence)mentioning

confidence: 99%

“…Using Lemma 1.13 in [2] (see also the proof of Theorem 3.5 in [3]), there exists a subsequence, still denoted by…”

Section: Theorem 2 (Convergence)mentioning

confidence: 99%

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Gradient Schemes for an Obstacle Problem

Alnashri

Droniou

2014

Springer Proceedings in Mathematics &Amp; Statistics

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The aim of this work is to adapt the gradient schemes, discretisations of weak variational formulations using independent approximations of functions and gradients, to obstacle problems modelled by linear and non-linear elliptic variational inequalities. It is highlighted in this paper that four properties which are coercivity, consistency, limit conformity and compactness are adequate to ensure the convergence of this scheme. Under some suitable assumptions, the error estimate for linear equations is also investigated.

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Finite Volume Methods: Foundation and Analysis

Barth

Herbin

Ohlberger

2017

Encyclopedia of Computational Mechanics Second Edition

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Finite volume methods are a class of discretization schemes that have proven highly successful in approximating the solution of a wide variety of conservation law systems. They are extensively used in fluid mechanics, porous media flow, meteorology, electromagnetics, models of biological processes, semi-conductor device simulation and many other engineering areas governed by conservative systems that can be written in integral control volume form.This article reviews elements of the foundation and analysis of modern finite volume methods.The primary advantages of these methods are numerical robustness through the obtention of discrete maximum (minimum) principles, applicability on very general unstructured meshes, and the intrinsic local conservation properties of the resulting schemes. Throughout this article, specific attention is given to scalar nonlinear hyperbolic conservation laws and the development of high order accurate schemes for discretizing them. A key tool in the design and analysis of finite volume schemes suitable for non-oscillatory discontinuity capturing is discrete maximum principle analysis. A number of building blocks used in the development of numerical schemes possessing local discrete maximum principles are reviewed in one and several space dimensions, e.g. monotone fluxes, E-fluxes, TVD discretization, nonoscillatory reconstruction, slope limiters, positive coefficient schemes, etc. When available, theoretical results concerning a priori and a posteriori error estimates are given. Further advanced topics are then considered such as high order time integration, discretization of dffision terms and the extension to systems of nonlinear conservation laws.

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Gradient Schemes: A Generic Framework for the Discretisation of Linear, Nonlinear and Nonlocal Elliptic and Parabolic Equations

Cited by 109 publications

References 25 publications

Gradient schemes for two‐phase flow in heterogeneous porous media and Richards equation

Gradient schemes for two‐phase flow in heterogeneous porous media and Richards equation

Gradient Schemes for an Obstacle Problem

Finite Volume Methods: Foundation and Analysis

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