Upwinding in finite element systems of differential forms Snorre H. Christiansen * Published as the Smale Prize Lecture in: Foundations of computational mathematics, Budapest 2011, London Mathematical Society Lecture Note Series, 403, Cambridge University Press, 2013. Abstract We provide a notion of finite element system, that enables the con- struction spaces of differential forms, which can be used for the numerical solution of variationally posed partial differential equations. Within this framework, we introduce a form of upwinding, with the aim of stabilizing methods for the purposes of computational fluid dynamics, in the vanish- ing viscosity regime. Foreword I am deeply honored to receive the first Stephen Smale prize from the Society for Foundations of Computational Mathematics. I want to thank the jury for deciding, in what I understand was a difficult weighting process, to tip the balance in my favor. The tiny margins that simi- larly enable the G¨ omb¨ oc to find its way to equilibrium, give me equal pleasure to contemplate. It’s a beautiful prize trophy. It is a great joy to receive a prize that celebrates the unity of mathemat- ics. I hope it will draw attention to the satisfaction there can be, in combining theoretical musings with potent applications. Differential geometry, which in- fuses most of my work, is a good example of a subject that defies perceived boundaries, equally appealing to craftsmen of various trades. As I was entering the subject, rumors that Smale could turn spheres inside out without pinching, were among the legends that gave it a sense of surprise and mystery. I also remember reading about Turing machines built on other rings than Z/2Z, which, together with parallelism and quantum computing, convinced me that the foundations of our subject were still in the making. Happy for the occasion provided by the FoCM conference, to meet the master, I was also a bit intimidated to learn that we have a common interest in discrete de Rham sequences. They are the topic of this paper. Many people have generously shared their insights and outlooks with me. I feel particularly indebted, mathematically as well as personally, to Jean-Claude * Centre of Mathematics for Applications & Department of Mathematics, University of Oslo, PO Box 1053 Blindern, NO–0363 Oslo, Norway 1