TBA

Although Green's theorem, currently considered one of the cornerstones of multivariate calculus, was published in 1828, its widespread introduction into calculus textbooks can be traced back to the first decades of the twentieth century, when vector calculus emerged as a slightly autonomous discipline. In addition, its contemporary version (and its demonstration), currently found in several calculus textbooks, is the result of some adaptations during its almost 200 years of life. Comparing some books and articles from this long period, I would like to discuss in this lecture the didactic adaptations, the editorial strategies and visual representations that shaped the theorem in its current form.

While scattering problems are posed on unbounded domains, volumetric discretizations typically require truncating the domain at a finite distance, closing the system with some sort of boundary condition.  These conditions typically suffer from some deficiency, such as perturbing the boundary value problem to be solved or changing the character of the operator so that the discrete system is difficult to solve with iterative methods.

We introduce a new technique for the Helmholtz problem, based on using the Green formula representation of the solution at the artificial boundary.  Finite element discretization of the resulting system gives optimal convergence estimates.  The resulting algebraic system can be solved effectively with a matrix-free GMRES implementation, preconditioned with the local part of the operator.  Extensions to the Morse-Ingard problem, a coupled system of pressure/temperature equations arising in modeling trace gas sensors, will also be given.

We investigate the long-time behaviour of solutions to a nonlocal partial differential equation on smooth Riemannian manifolds of bounded sectional curvature. The equation models self-collective behaviour with intrinsic interactions that are modeled by an interaction potential. Without the diffusion term, we consider attractive interaction potentials and establish sufficient conditions for a consensus state to form asymptotically. In addition, we quantify the approach to consensus, by deriving a convergence rate for the diameter of the solution’s support. With the diffusion term, the attractive interaction and the diffusion compete. We provide the conditions of the attractive interaction for each part to win.