227 citations · 335 across the 18 of their papers we have counts for
24 papers
Optimal error estimates for non-conforming approximations of linear parabolic problems with minimal regularity
J Droniou, R Eymard, T Gallouët +2
We consider a general linear parabolic problem with extended time boundary conditions (including initial value problems and periodic ones), and approximate it by the implicit Euler…
Space-time error estimates for approximations of linear parabolic problems with generalized time boundary conditions
Wolfgang Arendt, Isabelle Chalendar, Robert Eymard
We first give a general error estimate for the nonconforming approximation of a problem for which a Banach-Ne{\v c}as-Babu{\v s}ka (BNB) inequality holds. This framework covers par…
Convergence of the incremental projection method using conforming approximations
Robert Eymard, David Maltese
We prove the convergence of an incremental projection numerical scheme for the time-dependent incompressible Navier--Stokes equations, without any regularity assumption on the weak…
Extensions of derivations and symmetric operators
Wolfgang Arendt, Isabelle Chalendar, Robert Eymard
Given a densely defined skew-symmetric operators A 0 on a real or complex Hilbert space V , we parametrize all m-dissipative extensions in terms of contractions : H-$\rightarro…
Weighted p--Laplace approximation of linear and quasi-linear elliptic problems with measure data
Robert Eymard, David Maltese, Alain Prignet
We approximate the solution to some linear and degenerate quasi-linear problem involving a linear elliptic operator (like the semi-discrete in time implicit Euler approximation of…
Lions' representation theorem and applications
Wolfgang Arendt, Isabelle Chalendar, Robert Eymard
The Representation Theorem of Lions (RTL) is a version of the Lax--Milgram Theorem where completeness of one of the spaces is not complete. In this paper, RTL is deduced from an op…
Finite volume schemes and Lax-Wendroff consistency
R Eymard, T Gallouët, R Herbin +1
We present a (partial) historical summary of the mathematical analysis of finite differences and finite volumes methods, paying a special attention to the Lax-Richtmyer and Lax-Wen…
Exact pressure elimination for the Crouzeix-Raviart scheme applied to the Stokes and Navier-Stokes problems
Eric Chénier, Robert Eymard
We show that, using the Crouzeix-Raviart scheme, a cheap algebraic transformation, applied to the coupled velocity-pressure linear systems issued from the transient or steady Stoke…
Non-conforming finite elements on polytopal meshes
Jerome Droniou, Robert Eymard, Thierry Gallouet +1
In this work we present a generic framework for non-conforming finite elements on polytopal meshes, characterised by elements that can be generic polygons/polyhedra. We first prese…
Uniform-in-time convergence of numerical methods for non-linear degenerate parabolic equations
Jerome Droniou, Robert Eymard
Gradient schemes is a framework that enables the unified convergence analysis of many numerical methods for elliptic and parabolic partial differential equations: conforming and no…
A Class of Collocated Finite Volume Schemes for Incompressible Flow Problems
R. Eymard, R. Herbin, J. -C Latché +1
In this paper, we present a class of finite volume schemes for incompressible flow problems. The unknowns are collocated at the center of the control volumes, and the stability of…
Galerkin approximation of linear problems in Banach and Hilbert spaces
Wolfgang Arendt, Isabelle Chalendar, Robert Eymard
In this paper we study the conforming Galerkin approximation of the problem: find u U such that a(u, v) = <L, v> for all v V, where U and V are Hilbert or Banach spaces…
The gradient discretisation method for linear advection problems
Jérôme Droniou, Robert Eymard, T. Gallouët +1
We adapt the Gradient Discretisation Method (GDM), originally designed for elliptic and parabolic partial differential equations, to the case of a linear scalar hyperbolic equation…
High-order mass-lumped schemes for nonlinear degenerate elliptic equations
Jerome Droniou, Robert Eymard
We present and analyse a numerical framework for the approximation of nonlinear degenerate elliptic equations of the Stefan or porous medium types. This framework is based on piece…
A unified analysis of elliptic problems with various boundary conditions and their approximation
Jérôme Droniou, Robert Eymard, T. Gallouët +1
We design an abstract setting for the approximation in Banach spaces of operators acting in duality. A typical example are the gradient and divergence operators in Lebesgue--Sobole…
Unified convergence analysis of numerical schemes for a miscible displacement problem
Jérôme Droniou, Robert Eymard, Alain Prignet +1
This article performs a unified convergence analysis of a variety of numerical methods for a model of the miscible displacement of one incompressible fluid by another through a por…
Convergence in of weak solutions to perturbed doubly degenerate parabolic equations
Jérôme Droniou, Robert Eymard, Kyle S. Talbot
We study the behaviour of solutions to a class of nonlinear degenerate parabolic problems when the data are perturbed. The class includes the Richards equation, Stefan problem and…
Gradient schemes: generic tools for the numerical analysis of diffusion equations
Jerome Droniou, Robert Eymard, Raphaele Herbin
The gradient scheme framework is based on a small number of properties and encompasses a large number of numerical methods for diffusion models. We recall these properties and deve…
Singular limit of a two-phase flow problem in porous medium as the air viscosity tends to zero
Robert Eymard, Marie Henry, Danielle Hilhorst
In this paper we consider a two-phase flow problem in porous media and study its singular limit as the viscosity of the air tends to zero; more precisely, we prove the convergence…
A unified approach to Mimetic Finite Difference, Hybrid Finite Volume and Mixed Finite Volume methods
Jerome Droniou, Robert Eymard, Thierry Gallouët +1
We investigate the connections between several recent methods for the discretization of anisotropic heterogeneous diffusion operators on general grids. We prove that the Mimetic Fi…
A collocated finite volume scheme to solve free convection for general non-conforming grids
Eric Chénier, Robert Eymard, Raphaele Herbin
We present a new collocated numerical scheme for the approximation of the Navier-Stokes and energy equations under the Boussinesq assumption for general grids, using the velocity-p…
Discretisation of heterogeneous and anisotropic diffusion problems on general non-conforming meshes. SUSHI: a scheme using stabilisation and hybrid interfaces
Robert Eymard, Thierry Gallouët, Raphaele Herbin
A discretisation scheme for heterogeneous anisotropic diffusion problems on general meshes is developed and studied. The unknowns of this scheme are the values at the centre of the…
A mixed finite volume scheme for anisotropic diffusion problems on any grid
Jérôme Droniou, Robert Eymard
We present a new finite volume scheme for anisotropic heterogeneous diffusion problems on unstructured irregular grids, which simultaneously gives an approximation of the solution…
A cell-centred finite volume approximation for second order partial derivative operators with full matrix on unstructured meshes in any space dimension
Robert Eymard, Thierry Gallouët, Raphaèle Herbin
Finite volume methods for problems involving second order operators with full diffusion matrix can be used thanks to the definition of a discrete gradient for piecewise constant fu…