2019/05/06 by Jérôme Droniou, Droniou, Jerome, Kim-Ngan Le +1
Computer Science · Engineering · Mathematics · #65M12 #65M60 #76S05 #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Differential Equations and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.1905.01785
openalex publication_date 2019/05/06 · openalex created_date 2020/01/23 · openalex updated_date 2026/07/28
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 L2-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 ℙ1 finite elements and the Hybrid Mimetic Mixed method.