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

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-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.

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

“…The convergence orders are computed from the values of h, and the constant time steps have been taken proportional to h 2 . Note that in both cases, the proposed analytical solution is a strong solution for x 1 < t and x 1 > t and the Rankine-Hugoniot condition holds at the free boundary x 1 = t. It is therefore a weak solution to (2), extended to the case of non-homogeneous Dirichlet boundary conditions. These results confirm our uniformin-time convergence result (Theorem 4.1).…”

“…As mentioned in the introduction, we only sketch its proof and refer the reader to [2] for the details. …”

“…is actually a consequence of the other regularity properties on u, ζ (u), β (u) and of the equation, see [2].…”

“…Note that more complete proofs, as well as an entirely unified convergence analysis (not relying on previous convergence results) of Gradient Schemes for a more general and more non-linear model than (1), can be found in [2].…”

Uniform-in-Time Convergence of Numerical Schemes for a Two-Phase Discrete Fracture Model

“…The proof follows the ideas initially introduced in [1]. By the characterisation [2, Lemma 4.8] of uniform-in-time convergence, it suffices to prove that, for any sequence…”