Speaker: Dr Jérôme Droniou
Title: A unified convergence analysis framework for numerical approximations of diffusion equations
Date and time: Friday 18 August 2017, 3:00–4:00pm
Location: Building 8 Level 9 Room 66 (AGR) RMIT City campus
Abstract: Most partial differential equations modelling diffusion processes in engineering or physical problems are too complicated to be exactly solved. Numerical methods must be used to obtain quantitative information on their solutions. The role of mathematicians is twofold when dealing with such methods: design and analysis (each feeding the other). An abundance of different numerical methods can be found in the literature, and the usual way to proceed with their analysis is to perform one analysis for each method.
In this talk, we will present a recent framework – the Gradient Discretisation Method (GDM) – that was designed to unify all such analyses, and to pinpoint exactly which property of the numerical schemes are essential to ensure their convergence towards the proper continuous model. Understanding these properties also guides the design of new schemes. Surprisingly, only three such properties are actually required. By develop an abstract convergence analysis of numerical schemes, the GDM also enabled new results, including an answer to a long-standing conjecture on a popular numerical scheme in oil engineering.
Although most of the talk will be presented on diffusion equations, I will also discuss the potential application of the GDM to minimisation problems, when the functional involves the gradient of some quantity.
Bio: Jerome received his PhD from the university Aix-Marseille I in 2001. He was recruited in 2002 as Lecturer/Senior Lecturer at the University Montpellier II, and was then promoted Associate Prof. in 2006. In 2011 he moved to Australia and started working at Monash in 2012. His original fields of interest were in theoretical analysis of elliptic and parabolic partial differential equation, including in particular advection terms and measures as source terms. He gradually moved towards the numerical analysis of said equations, starting first with Finite Volume methods, before diversifying to other methods. Although he still maintains an activity in the theoretical analysis of these models, his current main center of interest is in the development of mathematical tools to facilitate the numerical analysis of complex elliptic and parabolic models, as encountered in real-world applications.