Uniform-in-time convergence of continuous and numerical approximations of doubly degenerate parabolic equations

Abstract

We consider a class of doubly degenerate parabolic equations

where $\beta$ and $\zeta$ are Lipschitz-continuous and non-decreasing, but may contain plateaux. The operator $-div\left(a\left(x,s,\cdot \right)\right)$ is a Leray–Lions operator acting on $W_0^{1,p}\left(\Omega \right)$, and $\nu$ is defined by ${\nu }^{\prime }={\beta }^{\prime }{\zeta }^{\prime }$. Particular cases of (1) are:

• Richards’ model of groundwater flow: ${\partial }_t\beta \left(u\right)-div\left(K\left(x,\beta \left(u\right)\right)\nabla u\right)=f$. Here, $u$ is the pressure and $\nu \left(u\right)=\beta \left(u\right)$ is the saturation.
• Stefan’s model of melting material: ${\partial }_{t}u-div\left(K\left(x,\zeta \left(u\right)\right)\nabla \zeta \left(u\right)\right)=f$. Here, $u$ is the internal energy and $\nu \left(u\right)=\zeta \left(u\right)$ is the temperature.
• Leray–Lions equations, with prototype the $p$-Laplace equation: ${\partial }_{t}u-div\left(|\nabla u{|}^{p-2}\nabla u\right)=f$. Such models are involved in non-Newtonian filtration. Here, $\nu \left(u\right)=u$.

I will show that a uniform analysis of all these models can be performed by a direct study of the generic model (1). I will present a stability result for the solutions of (1) under perturbations of the data $\beta ,\zeta ,a,f$. As a by-product, this establishes the existence of such solutions. The stability result is uniform-in-time and strong-in-space for $\nu \left(u\right)$, that is, in $C\left(\left[0,T\right];L^2\left(\Omega \right)\right)$. As demonstrated by the examples above, the value of $\nu \left(u\right)$ at a given time $T$ is the quantity of interest in practical applications. A uniform-in-time stability result on $\nu \left(u\right)$ is therefore particularly meaningful.

I will also consider the numerical approximations of (1) using the Gradient Schemes framework. This framework enables a unified analysis for many different numerical techniques (Finite Element, Mixed Finite Elements, Mimetic Finite Differences, etc.). I will demonstrate that the same strong convergence results can be obtain on the numerical approximation of $\nu \left(u\right)$.

For both theoretical and numerical analyses, the techniques used are based on compactness and compensated compactness results, and on energy estimates. These results and estimates are either continuous or discrete, depending on the analysis. The convergence is obtained without assuming non-physical regularity properties on the solution, which in particular allows for discontinuous permeabilities and conductivities (a natural kind of data in applications).

This is a joint work with Robert Eymard (University Paris-Est Marne-la-Vallée) and Kyle Talbot (Monash University).