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Computable reliable bounds for Poincaré–Friedrichs constants via Čech–de-Rham complexes

Martin Licht: Thursday, 25th April at 11:00 Abstract: We derive computable and reliable upper bounds for Poincaré–Friedrichs constants of classical Sobolev spaces and, more generally, Sobolev de-Rham complexes. The upper bounds are in terms of local Poincaré–Friedrichs constants over subdomains and the smallest singular value of a finite-dimensional operator that is easily assembled from the geometric setting. Thus we reduce the computational effort when computing the Poincaré–Friedrichs constant of finite de-Rham complexes, and we provide computable reliable bounds even for the original Sobolev de-Rham complex. The reduction to a finite-dimensional system uses diagram chasing within a Čech–de-Rham complex.

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Homogeneous multigrid for hybrid discretizations: application to HHO methods

Andreas Rupp： Tuesday, 2nd April at 14:00 Abstract: We prove the uniform convergence of the geometric multigrid V-cycle for hybrid high-order (HHO) and other discontinuous skeletal methods. Our results generalize previously established results for HDG methods, and our multigrid method uses standard smoothers and local solvers that are bounded, convergent, and consistent. We use a weak version of elliptic regularity in our proofs. Numerical experiments confirm our theoretical results.

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A localized orthogonal decomposition strategy for hybrid discontinuous Galerkin methods

Roland Maier： Thursday, 4th April at 11:00 Abstract: We formulate and analyze a multiscale method for an elliptic problem with an oscillatory coefficient based on a skeletal (hybrid) formulation. More precisely, we employ hybrid discontinuous Galerkin approaches and combine them with the localized orthogonal decomposition methodology to obtain a coarse-scale skeletal method that effectively includes fine-scale information. This work is a first step to reliably merge hybrid skeletal formulations and localized orthogonal decomposition and unite the advantages of both strategies. Numerical experiments are presented to illustrate the theoretical findings.

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Modeling some biological phenomena via the porous media approach

Zoubida Mghazli： Thursday, 18rd Jan at 14:00 Abstract Many biological systems can be modeled by the ”porous medium approach”, such as the diffusion of nutrients and other macromolecules through and in biological tissues. In this presentation, after a brief introduction to the ”porous medium approach”, we present some biological systems viewed through this approach. This will mainly concern the process of biodegradation of household waste, the Trichoderma fungi and the flow of water in the plant root.

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Trilogie sur les éléments finis

La trilogie Éléments Finis a été publiée dans la série Texts in Applied Mathematics de Springer en 2023. Co-écrite par Alexandre Ern de l’équipe SERENA et Jean-Luc Guermond de Texas A&M University, la trilogie comprend trois tomes, traitant de notions fondamentales d’approximation et d’interpolation (tome I), méthodes de Galerkin, EDP elliptiques et mixtes (tome II), et EDP de premier ordre et dépendantes du temps (tome III). Une caractéristique marquante est l’organisation de la trilogie en chapitres relativement petits (de 12 à 14 pages), facilitant ainsi son utilisation comme manuel pour l’enseignement des cours de niveau supérieur et facilitant également son utilisation comme référence pour les chercheurs.        

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