2015/06/22 by Jérôme Droniou, Jerome Droniou, Robert Eymard · 1 citation
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Compact space #Convergence (economics) #Degenerate energy levels #Elliptic partial differential equation #Finite element method #Numerical analysis #Numerical methods for differential equations #Parabolic partial differential equation #Partial differential equation #cs.NA #math.NA #msc:35K65 #msc:46N40 #msc:65M12
paper · pdf · doi:10.1007/s00211-015-0733-6
published as Numer. Math. 132 (4), pp. 721-766, 2016
openalex publication_date 2015/06/22 · crossref created 2015/06/22 · crossref issued 2015/06/23 · crossref published 2015/06/23 · crossref published-online 2015/06/23 · crossref published-print 2016/04/01 · openalex created_date 2016/06/24 · crossref deposited 2019/08/27 · arxiv created 2020/03/20 · arxiv updated 2020/03/23 · crossref indexed 2026/07/28 · openalex updated_date 2026/08/05
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.