arXiv subjects

Robert Eymard

Publications and source records attributed to Robert Eymard.

At least 19 recordsLinked to original sources

arXiv2026

The omnidirectional trace in H 1 ($\Omega$)

We first prove that all the functions in L 2 whose directional derivative is in L 2 have a directional trace on the boundary of any open bounded domain, without assumptions on its regularity. This enables us to define the omnidirectional trace of the elements of the Sobolev space H 1 ($\Omega$) for which there exists a function on the boundary that is almost everywhere equal, with respect to the directional measure, to the directional trace, regardless of the direction. The set of all these elements of H 1 ($\Omega$), denoted by H 1 tr ($\Omega$), is shown to be closed, and to always contain the closure in H 1 ($\Omega$) of the set C 0 ($\Omega$)$\cap$ H 1 ($\Omega$) (it is always equal to this set in the 1D case, and can be strictly greater in higher dimensions). The omnidirectional trace always satisfies an integration-by-parts formula, which combines the values of the trace on opposite points of the boundary. Examples show that this notion enables the resolution of variational problems involving the values at the boundary of the domain.

math.AP↗

arXiv2026

Convergence of a Control Volume Finite Element scheme for a cross-diffusion system modeling ion transport

An approximation of a system coupling the cross-diffusion of chemical species within a solvent, subjected to an electric field, is obtained through a control volume finite element (CVFE) scheme on general simplicial meshes in two or three space dimensions. The discrete unknowns of the numerical scheme are derived from the chemical potential of the species. The scheme is designed in order to fulfill entropy inequalities, yielding compactness properties for the discrete solutions and convergence to a weak solution of the continuous problem. Numerical illustrations of the convergence properties are provided in situations where diffusion of ionic species degenerates.

math.NA↗

arXiv2025

Trace operator on H 1 ($\Omega$) for general open bounded domains

In the case of any bounded open set $\Omega$ $\subset$ R d with boundary $\partial$$\Omega$, we first construct a directional trace in any direction $\theta$ of the unit sphere, for any u $\in$ L 2 ($\Omega$) whose the directional derivative $\partial$ $\theta$ u in the direction $\theta$ belongs to L 2 ($\Omega$). This directional trace is shown to belong to L 2 ($\partial$$\Omega$, $\mu$ $\theta$ ), where $\mu$ $\theta$ is a measure supported by the closure of all points of $\partial$$\Omega$ which are the extremity of an open segment directed by $\theta$, included in $\Omega$. This trace enables an integration by parts formula. We then show that the set H 1 tr ($\Omega$) containing the elements of H 1 ($\Omega$) whose the directional trace does not depend on $\theta$ is closed. It therefore contains the closure of H 1 ($\Omega$) $\cap$ C 0 ($\Omega$) in H 1 ($\Omega$). Examples where H 1 tr ($\Omega$) = H 1 ($\Omega$) and H 1 tr ($\Omega$) __ = H 1 ($\Omega$) are provided.

math.FA↗

arXiv2024

Optimal error bounds for the two point flux approximation finite volume scheme

We consider a finite volume scheme with two-point flux approximation (TPFA) to approximate a Laplace problem when the solution exhibits no more regularity than belonging to $H^1_0(Ω)$. We establish in this case some error bounds for both the solution and the approximation of the gradient component orthogonal to the mesh faces. This estimate is optimal, in the sense that the approximation error has the same order as that of the sum of the interpolation error and a conformity error. A numerical example illustrates the error estimate in the context of a solution with minimal regularity. This result is extended to evolution problems discretized via the implicit Euler scheme in an appendix.

math.NA↗

arXiv2023

Convergence of the incremental projection method using conforming approximations

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 solution. The velocity and the pressure are discretised in conforming spaces, whose the compatibility is ensured by the existence of an interpolator for regular functions which preserves approximate divergence free properties. Owing to a priori estimates, we get the existence and uniqueness of the discrete approximation. Compactness properties are then proved, relying on a Lions-like lemma for time translate estimates. It is then possible to show the convergence of the approximate solution to a weak solution of the problem. The construction of the interpolator is detailed in the case of the lowest degree Taylor-Hood finite element.

math.NA↗

arXiv2023

Space-time error estimates for approximations of linear parabolic problems with generalized time boundary conditions

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 parabolic problems with general conditions in time (initial value problems as well as periodic problems) under minimal regularity assumptions. We consider approximations by two types of space-time discretizations, both based on a conforming Galerkin method in space. The first one is the Euler $θ$--scheme. In this case, we show that the BNB inequality is always satisfied, and may require an extra condition on the time step for $θ$ $\le$ 1 2. The second one is the time discontinuous Galerkin method, where the BNB condition holds without any additional condition.

math.NA↗

arXiv2022

Extensions of derivations and symmetric operators

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 $Φ$ : H-$\rightarrow$ H + , where Hand H + are Hilbert spaces associated with a boundary quadruple. Such an extension generates a unitary C 0-group if and only if $Φ$ is a unitary operator. As corollary we obtain the parametrization of all selfadjoint extensions of a symmetric operator by unitary operators from Hto H +. Our results extend the theory of boundary triples initiated by von Neumann and developed by V. I. and M. L. Gorbachuk, J. Behrndt and M. Langer, S. A. Wegner and many others, in the sense that a boundary quadruple always exists (even if the defect indices are different in the symmetric case).

math.NA↗

arXiv2022

A new convergence proof for approximations of the Stefan problem

We consider the Stefan problem, firstly with regular data and secondly with irregular data. In both cases is given a proof for the convergence of an approximation obtained by regularising the problem. These proofs are based on weak formulations and on compactness results in some Sobolev spaces with negative exponents.

math.NA↗

arXiv2022

Weighted p--Laplace approximation of linear and quasi-linear elliptic problems with measure data

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 Richards and Stefan equations) with measure right-hand side and heterogeneous anisotropic diffusion matrix. This approximation is obtained through the addition of a weighted p--Laplace term. A well chosen diffeomorphism between R and (--1, 1) is used for the estimates of the approximated solution, and is involved in the above weight. We show that this approximation converges to a weak sense of the problem for general right-hand-side, and to the entropy solution in the case where the right-hand-side is in L 1 .

math.CA↗

arXiv2022

Lions' representation theorem and applications

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 operator-theoretical version on normed space. The main point of the paper is a theory of derivations, based on RTL, for which well-posedness is proved. One application concerns non-autonomous evolution equations with a new initial-value and a periodic boundary condition for the time variable.

math.FA↗

arXiv2021

Exact pressure elimination for the Crouzeix-Raviart scheme applied to the Stokes and Navier-Stokes problems

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 Stokes or Navier-Stokes problems, leads to a linear system only involving as many auxiliary variables as the velocity components. This linear system, which is symmetric positive definite in the case of the transient Stokes problem and symmetric invertible in the case of the steady Stokes problem, with the same stencil as that of the velocity matrix, provides the exact solution of the initial coupled linear system. Numerical results show the increase of performance when applying direct or iterative solvers to the resolution of these linear systems.

math.NA↗

arXiv2020

Non-conforming finite elements on polytopal meshes

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 present the functional framework on the example of a linear elliptic problem representing a single-phase flow in porous medium. This framework gathers a wide variety of possible non-conforming methods, and an error estimate is provided for this simple model. We then turn to the application of the functional framework to the case of a steady degenerate elliptic equation, for which a mass-lumping technique is required; here, this technique simply consists in using a different --piecewise constant-- function reconstruction from the chosen degrees of freedom. A convergence result is stated for this degenerate model. Then, we introduce a novel specific non-conforming method, dubbed Locally Enriched Polytopal Non-Conforming (LEPNC). These basis functions comprise functions dedicated to each face of the mesh (and associated with average values on these faces), together with functions spanning the local $\mathbb{P}^1$ space in each polytopal element. The analysis of the interpolation properties of these basis functions is provided, and mass-lumping techniques are presented. Numerical tests are presented to assess the efficiency and the accuracy of this method on various examples. Finally, we show that generic polytopal non-conforming methods, including the LEPNC, can be plugged into the gradient discretization method framework, which makes them amenable to all the error estimates and convergence results that were established in this framework for a variety of models.

math.NA↗

arXiv2020

Galerkin approximation of linear problems in Banach and Hilbert spaces

In this paper we study the conforming Galerkin approximation of the problem: find u $\in$ U such that a(u, v) = for all v $\in$ V, where U and V are Hilbert or Banach spaces, a is a continuous bilinear or sesquilinear form and L $\in$ V' a given data. The approximate solution is sought in a finite dimensional subspace of U, and test functions are taken in a finite dimensional subspace of V. We provide a necessary and sufficient condition on the form a for convergence of the Galerkin approximation, which is also equivalent to convergence of the Galerkin approximation for the adjoint problem. We also characterize the fact that U has a finite dimensional Schauder decomposition in terms of properties related to the Galerkin approximation. In the case of Hilbert spaces, we prove that the only bilinear or sesquilinear forms for which any Galerkin approximation converges (this property is called the universal Galerkin property) are the essentially coercive forms. In this case, a generalization of the Aubin-Nitsche Theorem leads to optimal a priori estimates in terms of regularity properties of the right-hand side L, as shown by several applications. Finally, a section entitled "Supplement" provides some consequences of our results for the approximation of saddle point problems.

math.NA↗

arXiv2020

Uniform-in-time convergence of numerical methods for non-linear degenerate parabolic equations

Gradient schemes is a framework that enables the unified convergence analysis of many numerical methods for elliptic and parabolic partial differential equations: conforming and non-conforming Finite Element, Mixed Finite Element and Finite Volume methods. We show here that this framework can be applied to a family of degenerate non-linear parabolic equations (which contain in particular the Richards', Stefan's and Leray--Lions' models), and we prove a uniform-in-time strong-in-space convergence result for the gradient scheme approximations of these equations. In order to establish this convergence, we develop several discrete compactness tools for numerical approximations of parabolic models, including a discontinuous Ascoli-Arzelà theorem and a uniform-in-time weak-in-space discrete Aubin-Simon theorem. The model's degeneracies, which occur both in the time and space derivatives, also requires us to develop a discrete compensated compactness result.

math.NA↗

arXiv2019

High-order mass-lumped schemes for nonlinear degenerate elliptic equations

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 piecewise constant approximations for the functions, which we show are essentially necessary to obtain convergence and error estimates. Convergence is established without regularity assumption on the solution. A detailed analysis is then performed to understand the design properties that enable a scheme, despite these piecewise constant approximations and the degeneracy of the model, to satisfy high-order error estimates if the solution is piecewise smooth. Numerical tests, based on continuous and discontinuous approximation methods, are provided on a variety of 1D and 2D problems, showing the influence on the convergence rate of the nature of the degeneracy and of the design choices.

math.NA↗

arXiv2019

The gradient discretisation method for linear advection problems

We adapt the Gradient Discretisation Method (GDM), originally designed for elliptic and parabolic partial differential equations, to the case of a linear scalar hyperbolic equations. This enables the simultaneous design and convergence analysis of various numerical schemes, corresponding to the methods known to be GDMs, such as finite elements (conforming or non-conforming, standard or mass-lumped), finite volumes on rectangular or simplicial grids, and other recent methods developed for general polytopal meshes. The scheme is of centred type, with added linear or non-linear numerical diffusion. We complement the convergence analysis with numerical tests based on the mass-lumped P1 conforming and non conforming finite element and on the hybrid finite volume method.

math.NA↗

arXiv2018

A unified analysis of elliptic problems with various boundary conditions and their approximation

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--Sobolev spaces on a bounded domain. We apply this abstract setting to the numerical approximation of Leray-Lions type problems, which include in particular linear diffusion. The main interest of the abstract setting is to provide a unified convergence analysis that simultaneously covers (i) all usual boundary conditions, (ii) several approximation methods. The considered approximations can be conforming, or not (that is, the approximation functions can belong to the energy space of the problem, or not), and include classical as well as recent numerical schemes. Convergence results and error estimates are given. We finally briefly show how the abstract setting can also be applied to other models, including flows in fractured medium, elasticity equations and diffusion equations on manifolds. A by-product of the analysis is an apparently novel result on the equivalence between general Poincar{\'e} inequalities and the surjectivity of the divergence operator in appropriate spaces.

math.NA↗

arXiv2017

Unified convergence analysis of numerical schemes for a miscible displacement problem

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 porous medium. The unified analysis is enabled through the framework of the gradient discretisation method for diffusion operators on generic grids. We use it to establish a novel convergence result in $L^\infty(0,T; L^2(\Omega))$ of the approximate concentration using minimal regularity assumptions on the solution to the continuous problem. The convection term in the concentration equation is discretised using a centred scheme. We present a variety of numerical tests from the literature, as well as a novel analytical test case. The performance of two schemes are compared on these tests; both are poor in the case of variable viscosity, small diffusion and medium to small time steps. We show that upstreaming is not a good option to recover stable and accurate solutions, and we propose a correction to recover stable and accurate schemes for all time steps and all ranges of diffusion.

math.NA↗