Gradient Discretisation Method

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. The core properties and analytical tools required to work within the GDM are stressed, and it is shown that scheme convergence can often be established by verifying a small number of properties. The scope of some of the featured techniques and results, such as time-space compactness theorems (discrete Aubin-Simon, discontinuous Ascoli-Arzela), goes beyond the GDM, making them potentially applicable to numerical schemes not (yet) known to fit into this framework. This monograph is intended for graduate students, researchers and experts in the field of the numerical analysis of partial differential equations.