OxPDE Lunchtime Seminar

Thu, 16/02 12:30	Reto Müller (Imperial College, London)	OxPDE Lunchtime Seminar	Gibson 1st Floor SR
	Geometric flows and their singularities
	In this talk, we first study the Mean Curvature Flow, an evolution equation for submanifolds of some Euclidean space. We review a famous monotonicity formula of Huisken and its application to classifying so-called Type I singularities. Then, we discuss the Ricci Flow, which might be seen as the intrinsic analog of the Mean Curvature Flow for abstract Riemannian manifolds. We explain how Huisken's classification of Type I singularities can be adopted to this intrinsic setting, using monotone quantities found by Perelman.

Thu, 01/03 12:30	François Murat (Université Paris VI)	OxPDE Lunchtime Seminar	Gibson 1st Floor SR
	Finite elements approximation of second order linear elliptic equations in divergence form with right-hand side in L¹
	In this lecture I will report on joint work with J. Casado-Díaz, T. Chacáon Rebollo, V. Girault and M.~Gómez Marmol which was published in Numerische Mathematik, vol. 105, (2007), pp. 337-510. We consider, in dimension $d\ge 2$ , the standard $P^1$ finite elements approximation of the second order linear elliptic equation in divergence form with coefficients in $L^\infty(\Omega)$ which generalizes Laplace's equation. We assume that the family of triangulations is regular and that it satisfies an hypothesis close to the classical hypothesis which implies the discrete maximum principle. When the right-hand side belongs to $L^1(\Omega)$ , we prove that the unique solution of the discrete problem converges in $W^{1,q}_0(\Omega)$ (for every $q$ with $1 \leq q$ < ${d \over d-1}$ ) to the unique renormalized solution of the problem. We obtain a weaker result when the right-hand side is a bounded Radon measure. In the case where the dimension is $d=2$ or $d=3$ and where the coefficients are smooth, we give an error estimate in $W^{1,q}_0(\Omega)$ when the right-hand side belongs to $L^r(\Omega)$ for some $r$ > $1$ .