arXiv subjects

Simon Lemaire

Publications and source records attributed to Simon Lemaire.

9 recordsLinked to original sources

arXiv2026

Equivalence of mixed and nonconforming methods on general polytopal partitions. Part I: Multiscale and projection methods

We study equivalence, in the context of a variable diffusion problem, between (conforming) mixed methods and (primal) nonconforming methods defined on potentially general polytopal partitions. In this first paper of a series of two, we focus on multiscale and projection methods. For multiscale methods, we establish the first-level equivalence between four different (oversampling-free) approaches, thereby broadening the results of [Chaumont-Frelet, Ern, Lemaire, Valentin; M2AN, 2022]. For projection methods, in turn, we provide a simple criterion (to be checked in practice) for primal/mixed well-posedness and equivalence to hold true. In the process, we also shed a new light on some self-stabilized hybrid methods. Part II of this work will address (general) polytopal element methods.

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arXiv2025

Commuting quasi-interpolators and Maxwell compactness for a polytopal de Rham complex

We establish Maxwell compactness results for the Discrete De Rham (DDR) polytopal complex: sequences in this polytopal complex with bounded discrete $\boldsymbol{H}(\mathbf{curl})$ (resp. discrete $\boldsymbol{H}(\mathrm{div})$) norm and orthogonal to discrete gradients (resp. discrete curls) have $L^2$-relatively compact potential reconstructions. The proof of these results hinges on the design of novel quasi-interpolators, that map the minimal-regularity de Rham spaces onto the discrete DDR spaces and form a commuting diagram. A full set of (primal and adjoint) consistency properties is established for these quasi-interpolators, which paves the way to convergence proofs, under minimal-regularity assumptions, of DDR schemes for partial differential equations based on the de Rham complex. Our analysis is performed with generic mixed boundary conditions, also covering the cases of no boundary conditions or fully homogeneous boundary conditions, and leverages recently introduced liftings from the DDR complex to the continuous de Rham complex.

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arXiv2025

Hybrid high-order approximations of div-curl systems on domains with general topology

We devise and analyze hybrid polyhedral methods of arbitrary order for the approximation of div-curl systems on three-dimensional domains featuring non-trivial topology. The div-curl systems we are interested in stem from magnetostatics, and can either be first-order (field formulation) or second-order (vector potential formulation). The well-posedness of the resulting discrete problems essentially hinges on recently established, topologically generic, hybrid versions of the (first and second) Weber inequalities. Our error analysis covers the case of regular solutions. Leveraging (co)homology computation techniques from the literature, we perform an in-depth numerical assessment of our approach, covering, in particular, the case of non-simply-connected domains.

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arXiv2024

Discrete Weber inequalities and related Maxwell compactness for hybrid spaces over polyhedral partitions of domains with general topology

We prove discrete versions of the first and second Weber inequalities on $\boldsymbol{H}(\mathbf{curl})\cap\boldsymbol{H}(\mathrm{div}_η)$-like hybrid spaces spanned by polynomials attached to the faces and to the cells of a polyhedral mesh. The proven hybrid Weber inequalities are optimal in the sense that (i) they are formulated in terms of $\boldsymbol{H}(\mathbf{curl})$- and $\boldsymbol{H}(\mathrm{div}_η)$-like hybrid semi-norms designed so as to embed optimally (polynomially) consistent face penalty terms, and (ii) they are valid for face polynomials in the smallest possible stability-compatible spaces. Our results are valid on domains with general, possibly non-trivial topology. In a second part we also prove, within a general topological setting, related discrete Maxwell compactness properties.

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arXiv2024

Structure preservation in high-order hybrid discretisations of potential-driven advection-diffusion: linear and nonlinear approaches

We are interested in the high-order approximation of anisotropic, potential-driven advection-diffusion models on general polytopal partitions. We study two hybrid schemes, both built upon the Hybrid High-Order technology. The first one hinges on exponential fitting and is linear, whereas the second is nonlinear. The existence of solutions is established for both schemes. Both schemes are also shown to possess a discrete entropy structure, ensuring that the long-time behaviour of discrete solutions mimics the PDE one. For the nonlinear scheme, the positivity of discrete solutions is a built-in feature. On the contrary, we display numerical evidence indicating that the linear scheme violates positivity, whatever the order. Finally, we verify numerically that the nonlinear scheme has optimal order of convergence, expected long-time behaviour, and that raising the polynomial degree results, also in the nonlinear case, in an efficiency gain.

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arXiv2022

Long-time behaviour of hybrid finite volume schemes for advection-diffusion equations: linear and nonlinear approaches

We are interested in the long-time behaviour of approximate solutions to heterogeneous and anisotropic linear advection-diffusion equations in the framework of hybrid finite volume (HFV) methods on general polygonal/polyhedral meshes. We consider two linear methods, as well as a new, nonlinear scheme, for which we prove the existence and the positivity of discrete solutions. We show that the discrete solutions to the three schemes converge exponentially fast in time towards the associated discrete steady-states. To illustrate our theoretical findings, we present some numerical simulations assessing long-time behaviour and positivity. We also compare the accuracy of the schemes on some numerical tests in the stationary case.

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arXiv2021

A discrete Weber inequality on three-dimensional hybrid spaces with application to the HHO approximation of magnetostatics

We prove a discrete version of the first Weber inequality on three-dimensional hybrid spaces spanned by vectors of polynomials attached to the elements and faces of a polyhedral mesh. We then introduce two Hybrid High-Order methods for the approximation of the magnetostatics model, in both its (first-order) field and (second-order) vector potential formulations. These methods are applicable on general polyhedral meshes, and allow for arbitrary orders of approximation. Leveraging the previously established discrete Weber inequality, we perform a comprehensive analysis of the two methods. We finally validate them on a set of test-cases.

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arXiv2017

A Hybrid High-Order method for highly oscillatory elliptic problems

We devise a Hybrid High-Order (HHO) method for highly oscillatory elliptic problems that is capable of handling general meshes. The method hinges on discrete unknowns that are polynomials attached to the faces and cells of a coarse mesh; those attached to the cells can be eliminated locally using static condensation. The main building ingredient is a reconstruction operator, local to each coarse cell, that maps onto a fine-scale space spanned by oscillatory basis functions. The present HHO method generalizes the ideas of some existing multiscale approaches, while providing the first complete analysis on general meshes. It also improves on those methods, taking advantage of the flexibility granted by the HHO framework. The method handles arbitrary orders of approximation $k\geq 0$. For face unknowns that are polynomials of degree $k$, we devise two versions of the method, depending on the polynomial degree $(k-1)$ or $k$ of the cell unknowns. We prove, in the case of periodic coefficients, an energy-error estimate of the form $\left(\varepsilon^{\frac{1}{2}}+H^{k+1}+(\varepsilon/H)^{\frac{1}{2}}\right)$, and we illustrate our theoretical findings on some test-cases.

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arXiv2017

On the best constant matrix approximating an oscillatory matrix-valued coefficient in divergence-form operators

We approximate an elliptic problem with oscillatory coefficients using a problem of the same type, but with constant coefficients. We deliberately take an engineering perspective, where the information on the oscillatory coefficients in the equation can be incomplete. A theoretical foundation of the approach in the limit of infinitely small oscillations of the coefficients is provided, using the classical theory of homogenization. We present a comprehensive study of the implementation aspects of our method, and a set of numerical tests and comparisons that show the potential practical interest of the approach. The approach detailed in this article improves on an earlier version briefly presented in [C. Le Bris, F. Legoll and K. Li, C. R. Acad. Sci. Paris 2013].

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