62 research outputs found
Introduction to discrete functional analysis techniques for the numerical study of diffusion equations with irregular data
We give an introduction to discrete functional analysis techniques for
stationary and transient diffusion equations. We show how these techniques are
used to establish the convergence of various numerical schemes without assuming
non-physical regularity on the data. For simplicity of exposure, we mostly
consider linear elliptic equations, and we briefly explain how these techniques
can be adapted and extended to non-linear time-dependent meaningful models
(Navier--Stokes equations, flows in porous media, etc.). These convergence
techniques rely on discrete Sobolev norms and the translation to the discrete
setting of functional analysis results
An HMM--ELLAM scheme on generic polygonal meshes for miscible incompressible flows in porous media
We design a numerical approximation of a system of partial differential
equations modelling the miscible displacement of a fluid by another in a porous
medium. The advective part of the system is discretised using a characteristic
method, and the diffusive parts by a finite volume method. The scheme is
applicable on generic (possibly non-conforming) meshes as encountered in
applications. The main features of our work are the reconstruction of a Darcy
velocity, from the discrete pressure fluxes, that enjoys a local consistency
property, an analysis of implementation issues faced when tracking, via the
characteristic method, distorted cells, and a new treatment of cells near the
injection well that accounts better for the conservativity of the injected
fluid
Numerical analysis for the pure Neumann control problem using the gradient discretisation method
The article discusses the gradient discretisation method (GDM) for
distributed optimal control problems governed by diffusion equation with pure
Neumann boundary condition. Using the GDM framework enables to develop an
analysis that directly applies to a wide range of numerical schemes, from
conforming and non-conforming finite elements, to mixed finite elements, to
finite volumes and mimetic finite differences methods. Optimal order error
estimates for state, adjoint and control variables for low order schemes are
derived under standard regularity assumptions. A novel projection relation
between the optimal control and the adjoint variable allows the proof of a
super-convergence result for post-processed control. Numerical experiments
performed using a modified active set strategy algorithm for conforming,
nonconforming and mimetic finite difference methods confirm the theoretical
rates of convergence
Gradient Schemes for Linear and Non-linear Elasticity Equations
The Gradient Scheme framework provides a unified analysis setting for many
different families of numerical methods for diffusion equations. We show in
this paper that the Gradient Scheme framework can be adapted to elasticity
equations, and provides error estimates for linear elasticity and convergence
results for non-linear elasticity. We also establish that several classical and
modern numerical methods for elasticity are embedded in the Gradient Scheme
framework, which allows us to obtain convergence results for these methods in
cases where the solution does not satisfy the full -regularity or for
non-linear models
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 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
Gradient schemes: a generic framework for the discretisation of linear, nonlinear and nonlocal elliptic and parabolic equations
International audienceGradient schemes are nonconforming methods written in discrete variational formulation and based on independent approximations of functions and gradients, using the same degrees of freedom. Previous works showed that several well-known methods fall in the framework of gradient schemes. Four properties, namely coercivity, consistency, limit-conformity and compactness, are shown in this paper to be sufficient to prove the convergence of gradient schemes for linear and nonlinear elliptic and parabolic problems, including the case of nonlocal operators arising for example in image processing. We also show that the Hybrid Mixed Mimetic family, which includes in particular the Mimetic Finite Difference schemes, may be seen as gradient schemes meeting these four properties, and therefore converges for the class of above mentioned problems
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
A unified approach to Mimetic Finite Difference, Hybrid Finite Volume and Mixed Finite Volume methods
We investigate the connections between several recent methods for the
discretization of anisotropic heterogeneous diffusion operators on general
grids. We prove that the Mimetic Finite Difference scheme, the Hybrid Finite
Volume scheme and the Mixed Finite Volume scheme are in fact identical up to
some slight generalizations. As a consequence, some of the mathematical results
obtained for each of the method (such as convergence properties or error
estimates) may be extended to the unified common framework. We then focus on
the relationships between this unified method and nonconforming Finite Element
schemes or Mixed Finite Element schemes, obtaining as a by-product an explicit
lifting operator close to the ones used in some theoretical studies of the
Mimetic Finite Difference scheme. We also show that for isotropic operators, on
particular meshes such as triangular meshes with acute angles, the unified
method boils down to the well-known efficient two-point flux Finite Volume
scheme
Some abstract error estimates of a finite volume scheme for a nonstationary heat equation on general nonconforming multidimensional spatial meshes
A general class of nonconforming meshes has been recently studied for stationary anisotropic heterogeneous diffusion problems by R. Eymard and coworkers. Thanks to these basic ideas developed for stationary problems, we derive a new discretization scheme in order to approximate the nonstationary heat problem. The unknowns of this scheme are the values at the centre of the control volumes, at some internal interfaces, and at the mesh points of the time discretization. Although the numerical scheme stems from the finite volume method, its formulation is based on the discrete version for the weak formulation defined for the heat problem. We derive error estimates for the solution in discrete norm, and an error estimate for an approximation of the gradient, in a general framework in which the discrete bilinear form is satisfying ellipticity. We prove in particular, that, when the discrete flux is calculated using a stabilized discrete gradient, the convergence order is h+k , where h (resp. k) is the mesh size of the spatial (resp. time) discretization. This estimate is valid under the regularity assumption that the exact solution is twice continuously differentiable in time and space. These error estimates are useful because they allow us to get error estimates for the approximations of the exact solution and its first derivatives
Schémas gradients appliqués à des problèmes elliptiques et paraboliques, linéaires et non-linéaires
The notion of gradient schemes, designed for linear and nonlinear elliptic and parabolic problems has the benefit of providing common convergence and error estimates results, which hold for a wide variety of numerical methods (finite element methods, nonconforming and mixed finite element methods, hybrid and mixed mimetic finite difference methods ...). Checking a minimal set of properties for a given numerical method suffices to prove that it belongs to the gradient schemes framework, and therefore that it is convergent on the different problems studied here. The study of the Stefan problem, the incompressible Stokes one and also the incompressible Navier-Stokes equations are presented in this thesis, where each one gets a convergence theorem set up with the gradient schemes framework. For Stokes and Navier-Stokes, we both provide the proof for the steady and the transient case dealing with some variational hypotheses which bring different convergence results. Finally, we also present four methods (Taylor-Hood, Crouzeix-Raviart, Marker-and-Cell, Hybrid Mixed Mimetic) for these two problems and we check that they enter in the gradient schemes frameworkLa notion de schémas gradients, conçue pour les équations elliptiques et paraboliques, linéaires et non-linéaires a l'avantage de fournir des résultats de convergence et d'estimations d'erreur valables pour de nombreuses familles de méthodes numériques (éléments finis conformes et non-conformes, éléments finis mixtes, différences finies ...). Vérifier un ensemble restreint de propriétés suffit pour prouver qu'une méthode numérique donnée rentre dans le cadre de travail des schémas gradients et donc qu'elle sera convergente sur les différents problèmes traités. L'étude du problème de Stefan, celle du problème de Stokes incompressible, ainsi que celle des équations de Navier-Stokes incompressibles sont présentées dans cette thèse, chacune présentant un théorème de convergence établi à l'aide des schémas gradients. Pour Stokes et Navier-Stokes, nous donnerons une preuve de convergence pour les cas stationnaires et transitoires en modifiant certaines hypothèses ce qui aura comme effet de trouver des résultats de convergence différents. Finalement, nous présentons également quatre méthodes (Taylor-Hood, Crouzeix-Raviart, Marker-and-Cell, Hybrid Mixed Mimetic) pour ces deux problèmes et nous vérifions qu'elles rentrent bien dans le cadre des schémas gradient
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