Scientific Computing Seminar
Abstract: We discuss the convergence properties of the so-called hybridizable discontinuous Galerkin (HDG) methods for second-order elliptic problems. The HDG methods are discontinuous Galerkin methods that can be implemented in a very efficient manner. They are also more precise than all the previously known discontinuous Galerkin methods for elliptic problems using spaces of equal degree for both the scalar unknown in its gradient. In this talk, we introduce these methods and explore, both numerically as well as theoreticlaly, their convergence properties for convection-diffusion problems in the diffusion-dominated regime. We also discuss numerical results for HDG methods for linear and nonlinear elasticity.
PDE Seminar
Abstract: A finite volume method is presented to discretize the Patlak-Keller-Segel (PKS) modelling chemosensitive movements. First, we prove existence and uniqueness of a numerical solution to the proposed scheme. Then, we give a priori estimates and establish a threshold on the initial mass, for which we show that the numerical approximation converges to the solution to the PKS system when the initial mass is lower than this threshold. Numerical simulations are performed to verify accuracy and the properties of the scheme. Finally, in the last section we investigate blow-up of the solution for large mass.