The gradient discretisation method for slow and fast diffusion porous media equations

TL;DR

This paper develops a unified convergence analysis for the gradient discretisation method applied to porous medium equations, including fast and slow diffusion, using discrete functional analysis techniques.

Contribution

It establishes strong $L^2$-convergence of approximate gradients and uniform-in-time convergence without regularity assumptions, applicable to various numerical schemes.

Findings

Convergence results hold for fast and slow diffusion regimes.

Numerical tests confirm theoretical convergence in practical methods.

Applicable to finite volume, finite element, and hybrid methods.

Abstract

The gradient discretisation method (GDM) is a generic framework for designing and analysing numerical schemes for diffusion models. In this paper, we study the GDM for the porous medium equation, including fast diffusion and slow diffusion models, and a concentration-dependent diffusion tensor. Using discrete functional analysis techniques, we establish a strong $L^2$-convergence of the approximate gradients and a uniform-in-time convergence for the approximate solution, without assuming non-physical regularity assumptions on the data or continuous solution. Being established in the generic GDM framework, these results apply to a variety of numerical methods, such as finite volume, (mass-lumped) finite elements, etc. The theoretical results are illustrated, in both fast and slow diffusion regimes, by numerical tests based on two methods that fit the GDM framework: mass-lumped conforming $\mathbb{P}_1$ finite elements and the Hybrid Mimetic Mixed method.