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This monograph presents the Gradient Discretisation Method (GDM), which is a unified convergence analysis framework for numerical methods for elliptic and parabolic partial differential equations. The results obtained by the GDM cover both stationary and transient models; error estimates are provided for linear (and some non-linear) equations, and convergence is established for a wide range of fully non-linear models (e.g. Leray-Lions equations and degenerate parabolic equations such as the Stefan or Richards models). The GDM applies to a diverse range of methods, both classical (conforming, non-conforming, mixed finite elements, discontinuous Galerkin) and modern (mimetic finite differences, hybrid and mixed finite volume, MPFA-O finite volume), some of which can be built on very general meshes.
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Daniele A. Di Pietro has been a Full Professor since 2012 and Deputy Director of Institut Montpelliérain Alexander Grothendieck at the University of Montpellier (France) since 2019. He is the author of two research monographs published by Springer and more than 80 scientific papers in refereed international journals or conference proceedings. His research fields include the development and analysis of advanced numerical methods for partial differential equations, with applications to fluid and solid mechanics and porous media. Over the course of his career, he has supervised ten PhD students and six postdoctoral fellows.
Jérôme Droniou was a Full Professor in France before moving to Monash university (Australia), where he has been an Associate Professor in the School of Mathematics since 2018, and head of the applied and computational section since 2019. He has authored two research monographs and more than 80 peer-reviewed articles and conference proceedings on theoretical and numerical analysis of partial differential equations. His current research interests revolve around the development of numerical methods for complex applications, and the design of mathematical tools to analyse the convergence of these methods. He has supervised a dozen PhD students and postdoctoral fellows in France and Australia.
