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Category: Uncategorized

Building (yet other) bridges between polytopal methods

Simon Lemaire: Thursday, 13th March 2025 at 10:30 Abstract: Within the last 20 years or so, a myriad of novel numerical approaches, capable of accommodating general polytopal meshes, have pop up in the literature. The main purpose of this talk is to tidy up the room, and to build connections, in the context of a model variable diffusion problem, between these different approaches. Our study will focus on skeletal methods. As opposed to plain-vanilla finite volume and discontinuous Galerkin discretizations, skeletal methods essentially attach degrees of freedom to the mesh skeleton. Our study will discriminate between primal and mixed formulations of the problem at hand. Somewhat unsurprisingly, we will see that, at the end of the day, all these approaches fall within only two distinct approximation paradigms.

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Computer-assisted proofs for nonlinear equations: how to turn a numerical simulation into a theorem.

Maxime Breden: Thursday 9th Nov at 11:00am Abstract: The goal of a posteriori validation methods is to get a quantitative and rigorous description of some specific solutions of nonlinear dynamical sys- tems, often ODEs or PDEs, based on numerical simulations. The general strategy consists in combining a priori and a posteriori error estimates, in- terval arithmetic, and a fixed point theorem applied to a quasi-Newton op- erator. Starting from a numerically computed approximate solution, one can then prove the existence of a true solution in a small and explicit neigh- borhood of the numerical approximation. I will first present the main ideas behind these techniques on a simple example, and then describe the results of a recent joint work with Jan Bouwe van den Berg and Ray Sheombarsing, in which we use these techniques to rigorously enclose solutions of some parabolic PDEs.

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