Computational and Applied Mathematics seminar

Abstract: The modified Hilbert transform arises naturally when considering a Galerkin-Bubnov variational formulation for the heat equation in anisotropic Sobolev spaces. In this talk, I will introduce the modified Hilbert transform and its main properties. I will explain how this operator leads to unique solvability of a variational formulation in anisotropic Sobolev spaces and to stability in the case of space-time tensor product discretization. Although the discrete inf-sup constant depending on the finite element mesh parameter initially suggests a reduced order of convergence due to the structure of Céa's lemma, optimal convergence is observed for a large class of functions.