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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Finite element trilogy

The Finite Element trilogy was published in the Series Texts in Applied Mathematics by Springer in 2023. Co-authored by Alexandre Ern from the SERENA Team and Jean-Luc Guermond from Texas A&M University, the trilogy comprises three volumes, dealing with fundamental notions of approximation and interpolation (volume I), Galerkin methods, elliptic and mixed PDEs (volume II), and first-order and time-dependent PDEs (volume III). A salient feature is the organization of the trilogy in relatively small chapters (from 12 to 14 pages), thereby facilitating its use as a textbook for teaching graduate level courses and also easing its use as a reference for researchers.        

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Computer-assisted proofs of radial solutions of elliptic systems on R^d

Olivier Hénot: Thursday 23rd Nov at 11:00 Abstract: The talk presents recent work on the rigorous computation of localized radial solutions of semilinear elliptic systems. While there are comprehensive results for scalar equations and some specific classes of elliptic systems, much less is known about these solutions in generic systems of nonlinear elliptic equations. These radial solutions are described by systems of non-autonomous ordinary differential equations. Using an appropriate Lyapunov-Perron operator, we rigorously enclose the centre-stable manifold, which contains the asymptotic behaviour of the profile. We then formulate, as a zero-finding problem, a shooting scheme from the set of initial conditions onto the invariant manifold. By means of a Newton-Kantorovich-type theorem, we obtain sufficient conditions to prove the existence and local uniqueness of a zero in the vicinity of a numerical approximation. We apply this method to prove ground state solutions for the Klein-Gordon equation on R^3, the Swift-Hohenberg equation on R^2, and a FitzHugh-Nagumo system on R^2.

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