Uniform-in-Time Convergence of Numerical Schemes for Richards’ and Stefan’s Models

2015

ANZIAMJ

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 of the data. For simplicity of exposure, we mostly consider linear elliptic equations, and we briefly explain how these techniques are adapted and extended to non-linear time-dependent meaningful models (Navier-Stokes equations, flows in porous media, etc.). These convergence techniques rely on Sobolev norms and discrete forms of functional analysis results. The discrete functional analysis tools presented here are versatile and applicable to a number of numerical methods and models.

Section: Convergencementioning

confidence: 99%

Section: Academic Examplementioning

confidence: 99%

Introduction to discrete functional analysis techniques for the numerical study of diffusion equations with irregular data

2015

ANZIAMJ

“…Note that the same inequality also holds with β and ζ swapped. Since these functions are non-decreasing, Inequality(24) follows from (23) and the similar inequality with β and ζ swapped. Since β is non-decreasing, β r is also non-decreasing on R β and therefore locally bounded on R β .…”

mentioning

confidence: 94%

“…We believe that the Discrete Functional Analysis results we establish in order to study the approximations of (1) -in particular the discrete compensated compactness theorem (Theorem 5.4) -could be critical to the numerical analysis of other degenerate or coupled models of physical importance. Note that the main results and their proofs have been sketched and illustrated by some numerical examples in [24], for a(x, ν(u), ∇ζ(u)) = ∇ζ(u) and β = Id or ζ = Id.…”

Section: Introduction 1motivationmentioning

confidence: 99%

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

Eymard

2015

Numer. Math.

Self Cite

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. 8050, 5 boulevard Descartes, Champs-sur-Marne 77454 Marne-la-Vallée Cedex 2, France. Robert.Eymard@univ-mlv.fr.2. The Stefan model [8], setting β(s) = s, ν = ζ, a(x, ν(u), ∇ζ(u)) = K(x, ζ(u))∇ζ (u), which arises in the study of a simplified heat diffusion process in a melting medium, 3. The p−Laplace problem, setting β(s) = ζ(s) = ν(s) = s and a(x, ν(u), ∇ζ(u)) = |∇u| p−2 ∇u, which is involved in the motion of glaciers [37] or flows of incompressible turbulent fluids through porous media [16].General Leray-Lions operators a(x, s, ξ) have growth, monotony and coercivity properties (see (2f)-(2h) below) which ensure that −div(a(x, w, ∇·)) maps W 1,p 0 (Ω) into W −1,p ′ (Ω), and thanks to which this differential operator is viewed as a generalisation of the p-Laplace operator.

“…It is noted that, for models based on linear operators as in (1.1) only three properties are sufficient to obtain the convergence of schemes for the corresponding PDEs. We refer the reader to [18,24,20,15,17,25,22] for more details.…”

Section: Introductionmentioning

confidence: 99%

Gradient schemes for the Signorini and the obstacle problems, and application to hybrid mimetic mixed methods

Alnashri

Computers & Mathematics with Applications

2016

Self Cite

Gradient schemes is a framework which enables the unified convergence analysis of many different methods -such as finite elements (conforming, non-conforming and mixed) and finite volumes methods -for 2 nd order diffusion equations. We show in this work that the gradient schemes framework can be extended to variational inequalities involving mixed Dirichlet, Neumann and Signorini boundary conditions. This extension allows us to provide error estimates for numerical approximations of such models, recovering known convergence rates for some methods, and establishing new convergence rates for schemes not previously studied for variational inequalities. The general framework we develop also enables us to design a new numerical method for the obstacle and Signorini problems, based on hybrid mimetic mixed schemes. We provide numerical results that demonstrate the accuracy of these schemes, and confirm our theoretical rates of convergence.