Upwinding in finite element systems of differential forms

We provide a notion of finite element system, that enables the construction spaces of differential forms, which can be used for the numerical solution of variationally posed partial difeerential equations. Within this framework, we introduce a form of upwinding, with the aim of stabilizing methods for the purposes of computational uid dynamics, in the vanishing viscosity regime. Published as the Smale Prize Lecture in: Foundations of computational mathematics, Budapest 2011, London Mathematical Society Lecture Note Series, 403, Cambridge University Press, 2013

A Two-Level Method for Mimetic Finite Difference Discretizations of Elliptic Problems

We propose and analyze a two-level method for mimetic finite difference approximations of second order elliptic boundary value problems. We prove that the two-level algorithm is uniformly convergent, i.e., the number of iterations needed to achieve convergence is uniformly bounded independently of the characteristic size of the underling partition. We also show that the resulting scheme provides a uniform preconditioner with respect to the number of degrees of freedom. Numerical results that validate the theory are also presented

Noncommutative Geometry and Gauge theories on AF algebras

Non-commutative geometry (NCG) is a mathematical discipline developed in the 1990s by Alain Connes. It is presented as a new generalization of usual geometry, both encompassing and going beyond the Riemannian framework, within a purely algebraic formalism. Like Riemannian geometry, NCG also has links with physics. Indeed, NCG provided a powerful framework for the reformulation of the Standard Model of Particle Physics (SMPP), taking into account General Relativity, in a single "geometric" representation, based on Non-Commutative Gauge Theories (NCGFT). Moreover, this accomplishment provides a convenient framework to study various possibilities to go beyond the SMPP, such as Grand Unified Theories (GUTs). This thesis intends to show an elegant method recently developed by Thierry Masson and myself, which proposes a general scheme to elaborate GUTs in the framework of NCGFTs. This concerns the study of NCGFTs based on approximately finite $C^*$-algebras (AF-algebras), using either derivations of the algebra or spectral triples to build up the underlying differential structure of the Gauge Theory. The inductive sequence defining the AF-algebra is used to allow the construction of a sequence of NCGFTs of Yang-Mills Higgs types, so that the rank $n+1$ can represent a grand unified theory of the rank $n$. The main advantage of this framework is that it controls, using appropriate conditions, the interaction of the degrees of freedom along the inductive sequence on the AF algebra. This suggests a way to obtain GUT-like models while offering many directions of theoretical investigation to go beyond the SMPP