Jérôme Droniou is Associate Professor at Monash University, Australia. His research focuses on elliptic and parabolic PDEs. He has published many papers on theoretical and numerical analysis of models with singularities or degeneracies, including convergence analysis of schemes without regularity assumptions on the data or solutions.Robert Eymard is professor of mathematics at Université Paris-Est Marne-la-Vallée. His research concerns  the design and analysis of numerical methods, mainly applied to fluid flows in porous media and incompressible Navier-Stokes equations.Thierry Gallouet is  professor at the University of Aix-Marseille. His research focuses on the analysis of partial differential equations and the approximation of their solutions by numerical schemes.  Cindy Guichard is assistant professor at Sorbonne Université. Her research is mainly focused on numerical methods for nonlinear fluid flows problems,  including  coupled  elliptic or parabolic equations  and  hyperbolic equations. Raphaèle Herbin is professor at the University of Aix-Marseille. She is a specialist of numerical schemes for partial differential equations, with application to incompressible and compressible fluid flows.

• Part I Elliptic problems.- Part II Parabolic problems.- Part III Examples of gradient discretisation methods.- Part IV Appendix.