The Gradient Discretization Method for Slow and Fast Diffusion Porous Media Equations
Jérôme Droniou, Kim Ngan Le
Abstract
The gradient discretization method (GDM) is a generic framework for designing and analyzing 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 nonphysical 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.