Accueil Articles publiés par Zhaonan Dong (Page 2)

Coarse spaces for non-hermitian and indefinite problems using two-level non-hermitian preconditioners

Emile Parolin: Thursday, 03rd July 2025 at 10:30 Abstract: Efficiently solving partial differential equation problems with highly heterogeneous coefficients is challenging due to the need for fine meshes, leading to large-scale and ill-conditioned linear systems. Domain decomposition methods address this by constructing efficient preconditioners that solve independent local problems in parallel. A key aspect to achieve scalability and robustness in these methods is the incorporation of a suitable coarse space. This presentation begins with an overview of one-level preconditioners, then introduces a novel algebraic construction of adaptive coarse spaces for two-level methods. The analysis is broadly applicable, encompassing non-hermitian and indefinite problems, as well as symmetric preconditioners like the additive Schwarz method and non-symmetric ones such as the restricted additive Schwarz method. It can also account for both exact and inexact subdomain solves. The coarse space is constructed by solving local generalized eigenproblems within each subdomain and applying a carefully chosen operator to the selected eigenvectors to obtain a local discrete solution. This is based on a joint work with Frédéric Nataf and Pierre-Henri Tournier.

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A discrete trace theory for non-conforming hybrid polytopal discretisation methods with application to analysis of BDDC preconditioners

Jai Tushar: Thursday, 10th July 2025 at 10:30 Abstract: Polytopal methods are a class of Finite Element Methods (FEMs) that have gained popularity in recent years due to their ability to relax conformity constraints on meshes. This flexibility makes them well-suited for handling complex geometries, adaptive mesh refinementand coarsening. The design of efficient, robust, scalable solvers for linear systems arising from these kind of discretisations is important to make them competitive with traditional methods. One family of such scalable preconditioners are non-overlapping domain decompositionmethods. The analysis of non-overlapping domain decomposition method-based solvers like BDDC relies on the exchange of information across inter-subdomain boundaries. It requires three main ingredients: a trace inequality, which implies that the restriction of functions to thesubdomain interface is stable; a lifting result, which lifts this restriction to the interior of the neighboring subdomain; and continuity of a face truncation operator on piecewise polynomial functions. The bound on this operator leads to a mesh-dependent logarithmic estimate. For conforming finite element methods, this is realized with the help of continuous trace theory. For non-conforming methods, such as polytopal methods, the continuous trace theory fails, since the trace of piecewise polynomial functions in L2(Ω) does not possess H1/2(∂Ω)regularity. The current state of the art to address this involves constructing an interpolant of a function on the interface (inter-subdomain boundary) onto a conforming finite elementspace and then applying the continuous trace theory. As a result, all the analysis for nonconforming spaces so far has been carried out on conforming simplicial/tetrahedral or quadrilateral/hexahedral meshes. In this talk, we will present a discrete trace theory for non-conforming polytopal methods. This theory is based entirely on the fully discrete hybrid spaces appearing in these methods. It hinges on the design of a novel discrete trace seminorm. For this seminorm, we establish discrete trace…

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Efficient numerical simulation of single-phase flow in three-dimensional fractured porous media

Daniel Zegarra Vasquez: Thursday, 15th May 2025 at 10:30 Abstract: Cette présentation est une répétition en vue de la soutenance de ma thèse qui aura lieu le mardi 27 mai 2025. Cette thèse porte sur les écoulements monophasiques dans des milieux poreux souterrains tridimensionnels, caractérisés par des fractures, des discontinuités étroites omniprésentes dans la matrice rocheuse. Dans ce travail, les fractures sont spécifiquement considérées comme des conduits privilégiés pour l’écoulement et sont modélisées via l’approche “Discrete Fracture Matrix” (DFM). Ce modèle conserve la tridimensionnalité de la matrice rocheuse et traite le réseau de fractures comme un objet de codimension un. L’écoulement est régi par des équations aux dérivées partielles (EDPs) de type Darcy couplées, assurant les échanges entre roche et fractures. Alors que la littérature se limite à des DFMs avec quelques milliers de fractures, cette thèse traite des modèles en comptant jusqu’à plusieurs centaines de milliers. L’objectif de ce travail est de concevoir, implémenter et analyser une méthode de simulation adaptée à ces DFMs. Le défi principal réside dans la résolution efficace des systèmes linéaires issus de la discrétisation des EDPs, pouvant atteindre plusieurs centaines de millions de degrés de liberté (DDL). Le chapitre [1] traite de l’analyse mathématique et numérique du système d’EDPs. Il fournit le cadre fonctionnel de la formulation variationnelle mixte et prouve l’existence et l’unicité de la solution. Les éléments finis mixtes sont utilisés pour poser le problème discret, pour lequel l’existence et l’unicité sont démontrées. Des estimations d’erreur a priori sont également établies. La formulation mixte-hybride équivalente (EFMH) conduit à un système linéaire creux, carré, symétrique et défini positif. La convergence numérique d’ordre 1 du schéma EFMH est validée par un test académique avec solution analytique. Le chapitre [2] propose une méthodologie pour évaluer les performances des solveurs linéaires appliqués au système issu de la…

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Geological storage of CO2: an example for the simulation of subsurface flow

Michel Kern: Thursday, 27th March 2025 at 10:30 Abstract: In this talk, I will present the main models used in the simulation of subsurface flow and why they are relevant for the storage of CO2 in deep geological formations. As an example, I will show results from a recent international benchmarking example. The talk is a rehearsal for my “Demi-heure de scicence” presentation on April 3rd.

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Hybrid high-order methods for the wave equation in first-order form

Rekha Khot: Thursday, 20th March 2025 at 10:30 Abstract: In this talk, we will discuss the approximation of the acoustic wave equation in its first-order Friedrichs formulation using hybrid high-order (HHO) methods, proposed and numerically investigated in [Burman-Duran-Ern, 2022]. We first look at energy-error estimates in the time-continuous setting and give several examples of interpolation operators: the classical one in the HHO literature based on L2 orthogonal projections and others from, or inspired from, the hybridizable discontinuous Galerkin (HDG) literature giving improved convergence rates on simplices. The time-discrete setting is based on explicit Runge-Kutta (ERK) schemes in time combined with HHO methods in space. In the fully discrete analysis, the key observation is that it becomes crucial to bound the consistency error in space by means of the stabilization seminorm only. We formulate three abstract properties (A1)-(A3) to lead the analysis. Our main result proves that, under suitable CFL conditions for second- and third-order ERK schemes, the energy error converges optimally in time and quasi-optimally in space, with optimal rates recovered on simplicial meshes. The abstract foundations of our analysis should facilitate its application to other nonconforming hybrid methods such as HDG and weak Galerkin (WG) methods.

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