442 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
Finite volume schemes for diffusion equations: introduction to and review of modern methods
We present Finite Volume methods for diffusion equations on generic meshes,
that received important coverage in the last decade or so. After introducing
the main ideas and construction principles of the methods, we review some
literature results, focusing on two important properties of schemes (discrete
versions of well-known properties of the continuous equation): coercivity and
minimum-maximum principles. Coercivity ensures the stability of the method as
well as its convergence under assumptions compatible with real-world
applications, whereas minimum-maximum principles are crucial in case of strong
anisotropy to obtain physically meaningful approximate solutions
A mixed finite volume scheme for anisotropic diffusion problems on any grid
We present a new finite volume scheme for anisotropic heterogeneous diffusion
problems on unstructured irregular grids, which simultaneously gives an
approximation of the solution and of its gradient. In the case of simplicial
meshes, the approximate solution is shown to converge to the continuous ones as
the size of the mesh tends to 0, and an error estimate is given. In the general
case, we propose a slightly modified scheme for which we again prove the
convergence, and give an error estimate. An easy implementation method is then
proposed, and the efficiency of the scheme is shown on various types of grids
-approximation properties of elliptic projectors on polynomial spaces, with application to the error analysis of a Hybrid High-Order discretisation of Leray-Lions problems
In this work we prove optimal -approximation estimates (with
) for elliptic projectors on local polynomial spaces. The
proof hinges on the classical Dupont--Scott approximation theory together with
two novel abstract lemmas: An approximation result for bounded projectors, and
an -boundedness result for -orthogonal projectors on polynomial
subspaces. The -approximation results have general applicability to
(standard or polytopal) numerical methods based on local polynomial spaces. As
an illustration, we use these -estimates to derive novel error
estimates for a Hybrid High-Order discretization of Leray--Lions elliptic
problems whose weak formulation is classically set in for
some . This kind of problems appears, e.g., in the modelling
of glacier motion, of incompressible turbulent flows, and in airfoil design.
Denoting by the meshsize, we prove that the approximation error measured in
a -like discrete norm scales as when
and as when .Comment: keywords: -approximation properties of elliptic projector on
polynomials, Hybrid High-Order methods, nonlinear elliptic equations,
-Laplacian, error estimate
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
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
A Hybrid High-Order method for Leray-Lions elliptic equations on general meshes
In this work, we develop and analyze a Hybrid High-Order (HHO) method for
steady non-linear Leray-Lions problems. The proposed method has several assets,
including the support for arbitrary approximation orders and general polytopal
meshes. This is achieved by combining two key ingredients devised at the local
level: a gradient reconstruction and a high-order stabilization term that
generalizes the one originally introduced in the linear case. The convergence
analysis is carried out using a compactness technique. Extending this technique
to HHO methods has prompted us to develop a set of discrete functional analysis
tools whose interest goes beyond the specific problem and method addressed in
this work: (direct and) reverse Lebesgue and Sobolev embeddings for local
polynomial spaces, -stability and -approximation properties for
-projectors on such spaces, and Sobolev embeddings for hybrid polynomial
spaces. Numerical tests are presented to validate the theoretical results for
the original method and variants thereof
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