Abstract
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-Arzel\`a 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.
Highlights
1.1 MotivationThe following generic nonlinear parabolic model∂tβ(u) − div (a(x, ν(u), ∇ζ(u))) = f in Ω × (0, T ), β(u)(x, 0) = β(uini)(x) in Ω, (1)ζ(u) = 0 on ∂Ω × (0, T ), where β and ζ are non-decreasing, ν is such that ν′ = β′ζ′ and a is a Leray–Lions operator, arises in various frameworks
The Richards model, setting ζ(s) = s, ν = β and a(x, ν(u), ∇ζ(u)) = K(x, β(u))∇u, which describes the flow of water in a heterogeneous anisotropic underground medium, ∗School of Mathematical Sciences, Monash University, Victoria 3800, Australia. jerome.droniou@monash.edu. †Universite Paris-Est, Laboratoire d’Analyse et de Mathematiques Appliquees, UMR 8050, 5 boulevard Descartes, Champs-sur-Marne 77454 Marne-la-Vallee Cedex 2, France
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]
Summary
In [43] the authors study the stability and convergence properties of linearised implicit methods for the time discretization of nonlinear parabolic equations in the general framework of Hilbert spaces. The usual way to obtain pointwise-in-time approximation results for numerical schemes is to prove estimates in L∞(0, T ; L2(Ω)) on u − u, where u is the approximated solution Establishing such error estimates is only feasible when uniqueness of the solution u to (1) can be proved, which is the case for Richards’ and Stefan’s problems (with K only depending on x), but not for more complex non-linear parabolic problems as (1) or even p-Laplace problems. The framework we choose is that of gradient schemes, which has the double benefit of covering a vast number of numerical methods, and of having already been studied for many models – elliptic, parabolic, linear or non-linear, possibly degenerate, etc.
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