Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes SUSHI: a scheme using stabilization and hybrid interfaces

Eymard, Robert; Gallouët, Thierry; Herbin, Raphaèle

doi:10.1093/imanum/drn084

Cited by 290 publications

(532 citation statements)

References 26 publications

(65 reference statements)

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“…Then, following [5], a discrete gradient is defined on each cone K σ which only depends on v K and v σ ′ for σ ′ ∈ E K . This gradient is exact on linear functions and satisfies a weak convergence property.…”

Section: Brief Reminder Of the Hybrid Finite Volume Discretization Ofmentioning

confidence: 99%

“…This gradient is exact on linear functions and satisfies a weak convergence property. According to [5], it can be written…”

Section: Brief Reminder Of the Hybrid Finite Volume Discretization Ofmentioning

confidence: 99%

“…Our discretization is based on the family of Hybrid Finite Volume schemes introduced for diffusion problems on general meshes in [5] and also closely related to Mimetic Finite Difference schemes [6] as shown in [4]. The degrees of freedom are defined by the displacement vector u σ at the center of gravity of each face σ of the mesh as well as the displacement vector u K at a given point x K of each cell K of the mesh.…”

Section: Introductionmentioning

confidence: 99%

“…The degrees of freedom are defined by the displacement vector u σ at the center of gravity of each face σ of the mesh as well as the displacement vector u K at a given point x K of each cell K of the mesh. Following [5], a piecewise constant gradient is built and can be readily used to define the discrete variational formulation which mimics the continuous variational formulation for the displacement vector field. In order to stabilize this formulation and to reduce the degrees of freedom, the tangential components of the displacement are interpolated in terms of the neighbouring normal components.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Hybrid Finite Volume Discretization of Linear Elasticity Models on General Meshes

Pietro¹,

Eymard²,

Lemaire³

et al. 2011

Finite Volumes for Complex Applications VI Problems &Amp; Perspectives

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This paper presents a new discretization scheme for linear elasticity models using only one degree of freedom per face corresponding to the normal component of the displacement. The scheme is based on a piecewise constant gradient construction and a discrete variational formulation for the displacement field. The tangential components of the displacement field are eliminated using a second order linear interpolation. Our main motivation is the coupling of geomechanical models and porous media flows arising in reservoir or CO2 storage simulations. Our scheme guarantees by construction the compatibility condition between the displacement discretization and the usual cell centered finite volume discretization of the Darcy flow model. In addition it applies on general meshes possibly non conforming such as Corner Point Geometries commonly used in reservoir and CO2 storage simulations.

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Section: Brief Reminder Of the Hybrid Finite Volume Discretization Ofmentioning

confidence: 99%

“…This gradient is exact on linear functions and satisfies a weak convergence property. According to [5], it can be written…”

Section: Brief Reminder Of the Hybrid Finite Volume Discretization Ofmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Hybrid Finite Volume Discretization of Linear Elasticity Models on General Meshes

Pietro¹,

Eymard²,

Lemaire³

et al. 2011

Finite Volumes for Complex Applications VI Problems &Amp; Perspectives

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show abstract

“…We also refer to [13] for complementary results. The proof of Lemma 3, which is adapted from the proof in [6], is given here for completeness of the presentation.…”

Section: 3mentioning

confidence: 99%

A finite-volume scheme for the multidimensional quantum drift-diffusion model for semiconductors

Chainais-Hillairet

Gisclon

Jüngel

2010

Numer. Methods Partial Differential Eq.

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Abstract. A finite-volume scheme for the stationary unipolar quantum drift-diffusion equations for semiconductors in several space dimensions is analyzed. The model consists of a fourth-order elliptic equation for the electron density, coupled to the Poisson equation for the electrostatic potential, with mixed Dirichlet-Neumann boundary conditions. The numerical scheme is based on a Scharfetter-Gummel type reformulation of the equations. The existence of a sequence of solutions to the discrete problem and its numerical convergence to a solution to the continuous model are shown. Moreover, some numerical examples in two space dimensions are presented.

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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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Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes SUSHI: a scheme using stabilization and hybrid interfaces

Cited by 290 publications

References 26 publications

Hybrid Finite Volume Discretization of Linear Elasticity Models on General Meshes

Hybrid Finite Volume Discretization of Linear Elasticity Models on General Meshes

A finite-volume scheme for the multidimensional quantum drift-diffusion model for semiconductors

Finite Volume Methods: Foundation and Analysis

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