Jérôme Droniou Knihy

The book introduces Hybrid High-Order methods for addressing diffusive problems in computational mechanics, highlighting their innovative features compared to traditional numerical methods. Key aspects include the ability to utilize polytopal meshes with non-star-shaped elements, support for various approximation orders across dimensions, and improved physical compliance. Additionally, these methods offer reduced computational costs through compact stencils and static condensation, making them a significant advancement in the field of partial differential equations.

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