2025‌​‌Activity reportProject-TeamRAPSODI​​

3.1 Axis​‌ 1: Study of the​​ structural properties of continuous​​​‌ models

A well-behaved continuous​ (PDE based) model is​‌ a requisite to a​​ reliable numerical approximation. The​​​‌ fine understanding of a​ continuous model is indeed​‌ a prior to the​​ design of a numerical​​​‌ scheme in order to​ identify the structural properties​‌ to be preserved at​​ the discrete level. Further,​​​‌ the mathematical analysis carried​ out on the continuous​‌ model often paves the​​ way for the theoretical​​​‌ foundations of the numerical​ methods, hence our strong​‌ interest in this first​​ research axis. Analyzing time-dependent​​ PDE systems relying on​​​‌ their mathematical (Hamiltonian, gradient‌ flow, ...) structure is‌​‌ very natural and unsurprisingly​​ not new. However, the​​​‌ interest in this domain‌ strongly increased in the‌​‌ last two decades. The​​ focus of the RAPSODI​​​‌ project-team encompasses kinetic models,‌ fluid mechanics, electromagnetism, multicomponent‌​‌ systems and applied calculus​​ of variations. While carrying​​​‌ out theoretical analyses, we‌ bear in mind that‌​‌ our results should be​​ transposed to the discrete​​​‌ setting.

3.2 Axis 2: Design​‌ and analysis of structure-preserving​​ numerical methods

The more​​​‌ complex is the PDE​ system to be discretized,​‌ the smaller is the​​ chance that a naive​​​‌ discretization preserves at the​ discrete level its mathematical​‌ structure. Since this structure​​ was the cornerstone of​​​‌ the mathematical analysis (typically​ by providing the well-posedness​‌ and the stability of​​ the continuous system under​​​‌ consideration), its preservation at​ the discrete level will​‌ be key to get​​ well-behaved and theoretically certified​​​‌ numerical methods. Our team​ concentrates a large part​‌ of its research effort​​ on the design of​​​‌ provably convergent numerical methods,​ either based on the​‌ popular two-point flux approximation​​ finite volume method, or​​​‌ on more flexible yet​ less natural structure preserving​‌ methods. We also pay​​ attention to the fact​​​‌ that our schemes are​ robust with respect to​‌ the parameters and in​​ the long time limit.​​​‌ The numerical approximation of​ complex (inhomogeneous or with​‌ low compressibility) flows is​​ one of the topics​​​‌ we address.

3.3 Axis 3: Computational‌ optimization of the numerical‌​‌ methods

Good numerical methods​​ shall enjoy strong theoretical​​​‌ foundations allowing to guarantee‌ their behavior in very‌​‌ general situations. They shall​​ also be highly efficient​​​‌ from a computational point‌ of view, so that‌​‌ they can be used​​ in practice for solving​​​‌ real-world problems.

To increase‌ the efficiency and applicability‌​‌ of our methods, we​​ work on three tracks.​​​‌ First, we aim at‌ building a unified and‌​‌ optimized software platform to​​ implement, test and diffuse​​​‌ our numerical methods. Second,‌ since our methods often‌​‌ yield nonlinear systems, the​​ improvement of the nonlinear​​​‌ solvers is also key‌ to increase the efficiency‌​‌ of our methods. Finally,​​ for high-dimensional models such​​​‌ as kinetic and particle‌ systems, the curse of‌​‌ dimensionality makes numerical computations​​ realistically feasible only if​​​‌ specific computationally efficient numerical‌ strategies are deployed.

4.1 Subsurface​‌ CO2 storage

The capture​​ and storage in the​​​‌ subsurface of carbon dioxide​ is commonly acknowledged as​‌ a promising solution to​​ mitigate the emission of​​​‌ greenhouse gas from localized​ production sites, as for​‌ instance cement plants. The​​ safety assessment of the​​​‌ subsurface sequestration sites requires​ advanced numerical tools building​‌ on theoretically certified numerical​​ models and algorithms which​​​‌ remain valid in the​ long time limit. Among​‌ the numerous difficulties encountered​​ in such a setting,​​​‌ let us mention the​ high level of coupling​‌ between the mechanics of​​ the (solid) porous matrix​​​‌ 121, the multiphase​ and multicomponent character of​‌ the fluid which flows​​ therein 88, and​​​‌ chemical reactions with a​ wide range of characteristic​‌ times 138. Despite​​ important differences (at the​​​‌ level of chemistry especially),​ similar problems occur in​‌ the emerging topic of​​ dihydrogen subsurface storage, or​​​‌ native dihydrogen (generated by​ the corrosion of iron​‌ in an aqueous environment)​​ migration in the context​​ of nuclear waste repository​​​‌ management.

Together with colleagues‌ from applied research institutes‌​‌ and academics, the RAPSODI​​ project-team contributes to the​​​‌ derivation of so-called thermodynamically‌ consistent models, the‌​‌ stability of which in​​ the long-time limit being​​​‌ guaranteed by the second‌ principle. The team also‌​‌ contributes to the design​​ and the analysis of​​​‌ numerical schemes for multiphase‌ and multicomponent flows in‌​‌ complex geometries (possibly allowing​​ for general meshes), and​​​‌ to the design of‌ fast and robust solvers‌​‌ for chemical equilibria.

4.2​​ Material sciences

The team​​​‌ is interested in the‌ theoretical and numerical analysis‌​‌ of mathematical models describing​​ the degradation of materials,​​​‌ as concrete carbonation and‌ corrosion. The study of‌​‌ such models is an​​ important environmental and industrial​​​‌ issue. Atmospheric carbonation degrades‌ reinforced concretes and limits‌​‌ the lifetime of civil​​ engineering structures. Corrosion phenomena​​​‌ issues occur for instance‌ in the reliability of‌​‌ nuclear power plants and​​ the nuclear waste repository.​​​‌ The study of the‌ long time evolution of‌​‌ these phenomena is of​​ course fundamental in order​​​‌ to predict the lifetime‌ of the structures.

4.3‌​‌ Complex fluid flows

The​​ team is interested in​​​‌ numerical methods for the‌ simulation of systems of‌​‌ PDEs describing complex flows,​​ like for instance mixture​​​‌ flows, granular gases, rarefied‌ gases, or quantum fluids.‌​‌

Variable-density, low-Mach flows have​​ been widely studied in​​​‌ the recent literature because‌ of their applicability in‌​‌ various phenomena such as​​ flows in high-temperature gas​​​‌ reactors, meteorological flows, flows‌ with convective and/or conductive‌​‌ heat transfer or combustion​​ processes. In such cases,​​​‌ the resolution of the‌ full compressible Navier–Stokes system‌​‌ is not adapted, because​​ of the sound waves'​​​‌ speed. The Boussinesq incompressible‌ model is not a‌​‌ better alternative for such​​ low-speed phenomena, because the​​​‌ compressibility effects cannot be‌ totally cancelled due to‌​‌ large variations of temperature​​ and density. Consequently, some​​​‌ models have been formally‌ derived, leading to the‌​‌ filtering of the acoustic​​ waves by the use​​​‌ of some formal asymptotic‌ expansions and two families‌​‌ of methods have been​​ developed in the literature​​​‌ in order to compute‌ these flows. We are‌​‌ interested in particular in​​ the so-called pressure-based methods,​​​‌ which are more robust‌ than density-based solvers, although‌​‌ their range of validity​​ is in general more​​​‌ limited.

Quantum models can‌ be used to describe‌​‌ superfluids, quantum semiconductors, weakly​​ interacting Bose gases, or​​​‌ quantum trajectories of Bohmian‌ mechanics. They have attracted‌​‌ considerable attention in the​​ last decades, due in​​​‌ particular to the development‌ of nanotechnology applications. To‌​‌ describe quantum phenomena, there​​ exists a large variety​​​‌ of models. In particular,‌ there exist three different‌​‌ levels of description: microscopic,​​ mesoscopic, and macroscopic. The​​​‌ quantum Navier–Stokes equations deal‌ with a macroscopic description‌​‌ in which the quantum​​ effects are taken into​​​‌ account through a third-order‌ term called the quantum‌​‌ Bohm potential. This Bohm​​ potential arises from the​​​‌ fluid dynamical formulation of‌ the single-state Schrödinger equation.‌​‌ The nonlocality of quantum​​ mechanics is approximated by​​​‌ the fact that the‌ equations of state do‌​‌ not only depend on​​​‌ the particle density but​ also on its gradient.​‌ These equations were employed​​ to model field emissions​​​‌ from metals and steady-state​ tunneling in metal-insulator-metal structures,​‌ and to simulate ultra-small​​ semiconductor devices.

4.4 Electromagnetism​​​‌ for non-invasive control

The​ RAPSODI project-team works on​‌ the development of high-order​​ polyhedral methods for electromagnetism.​​​‌ A well-known specificity in​ electromagnetism is that topology​‌ plays a crucial role​​ in the well-posedness of​​​‌ the models. Dedicated analysis​ tools must then be​‌ deployed for their study​​ 82. We are​​​‌ interested in the devising​ of HHO methods in​‌ the curl/curl setting 118​​. The mathematical analysis​​​‌ of HHO methods in​ this setting is particularly​‌ involved, as most of​​ the needed discrete functional​​​‌ analysis tools are currently​ lacking. The design of​‌ (efficient and robust) multilevel​​ linear solvers for statically​​​‌ condensed HHO approximations of​ electromagnetic models, as well​‌ as of computable (reliable​​ and locally efficient) a​​​‌ posteriori error estimators on​ polyhedral cells are two​‌ other, still largely unexplored,​​ aspects we are interested​​​‌ in. Our main target​ application, in the framework​‌ of a recently initiated​​ collaboration with EDF (see​​​‌ Section 8.2), is​ the simulation of eddy​‌ current testing (ECT). ECT​​ is used by EDF​​​‌ as a non-invasive control​ technique to assess the​‌ integrity of heat exchanger​​ tubes in nuclear plants.​​​‌ The forward problem consists​ in solving the time-harmonic​‌ 3D Maxwell's equations in​​ domains featuring more or​​​‌ less complex flaws. The​ use of high-order polyhedral​‌ methods is expected to​​ yield improvements on two​​​‌ aspects. First, the support​ of polyhedral cells is​‌ expected to ease the​​ full meshing process, and​​​‌ in particular enable to​ account for defects with​‌ complex geometries/topologies. Second, the​​ increase in the approximation​​​‌ order is expected to​ yield a reduction of​‌ the noise on the​​ computed control signal based​​​‌ on which the presence​ of a defect is​‌ inferred.

4.5 Large population​​ models in epidemiology

Developing​​​‌ mathematical models to describe​ how infectious pathogens spread​‌ in animal populations is​​ an essential step to​​​‌ identify the main biological​ mechanisms or environmental factors​‌ which contribute to the​​ emergence of epidemics. When​​​‌ describing a pathogen spread​ at large scales, it​‌ is often relevant to​​ model the distributions of​​​‌ hosts (which may represent​ individuals, animals or herds,​‌ for example) as spatially​​ varying densities, and to​​​‌ model the pathogen dynamics​ via PDEs which describe​‌ the combined effect of​​ neighbourhood interactions, large scale​​​‌ population dynamics, and environmental​ factors.

The team initiated​‌ a collaboration with INRAE​​ (National Research Institute on​​​‌ Agriculture and Environment) on​ the modeling of different​‌ types of pathogen spread​​ mechanisms in such continous​​​‌ models. A first focus​ of this collaboration is​‌ on neighbourhood interactions, which​​ are a dominant factor​​​‌ in the infection dynamics​ of many pathogens (a​‌ specific example is the​​ Bovine Viral Diarrhea Virus,​​​‌ an endemic disease present​ worldwide among industrial cattle​‌ herds, and causing annually​​ large economic losses), and​​​‌ which at large scales​ may be described via​‌ non-local (convolution) terms. The​​ analysis of the resulting​​ model requires the development​​​‌ of dedicated mathematical tools,‌ and will lead to‌​‌ a better understanding of​​ the influence of neighborhood​​​‌ interactions on the spatial‌ features of the epidemic‌​‌ dynamics. A second focus​​ is on arboviruses, which​​​‌ are pathogens transmitted to‌ mammals by the bite‌​‌ of arthropod vectors, mainly​​ mosquitoes, and causing diseases​​​‌ such as the Zika‌ virus, the Rift Valley‌​‌ fever and the West​​ Nile virus. In this​​​‌ case, the aim is‌ to produce a comprehensive‌​‌ description of the viral​​ dynamics both in the​​​‌ vectors and in the‌ hosts as well as‌​‌ to model their interaction​​ in space (via diffusion​​​‌ processes and nonlocal interactions),‌ which will then be‌​‌ used to identify the​​ main mechanisms driving the​​​‌ virus spread.

4.6 Particle‌ accelerators

Relativistic electron bunches‌​‌ are used in storage​​ rings to produce intense​​​‌ radiation in various ranges‌ of frequencies. The dynamics‌​‌ of these bunches is​​ nonlinear because of interactions​​​‌ between electrons in the‌ bunch. Moreover, these interactions‌​‌ occur in an asymmetric​​ fashion because of the​​​‌ relativistic nature of the‌ dynamics. The stability properties‌​‌ of the bunch (and​​ thus the long-time properties​​​‌ of the dynamics) have‌ been shown, both theoretically‌​‌ and experimentally, to be​​ crucial in the understanding​​​‌ of the intensity of‌ the emitted radiation 98‌​‌, 126.

From​​ a mathematical point of​​​‌ view, the dynamics of‌ the electron density in‌​‌ the phase space can​​ be described by a​​​‌ Vlasov–Fokker–Planck type equation 98‌ with a well suited‌​‌ mean-field self-interaction term 142​​. The mathematical understanding​​​‌ of the long time‌ behavior of solutions to‌​‌ this equation and the​​ design of adapted numerical​​​‌ schemes constitutes a challenging‌ and physically important problem.‌​‌

6.1 Latest software developments​‌

7.1​​ Modeling, analysis and numerical​​​‌ simulation of multi-component systems‌

Participants: Alain Blaustein,‌​‌ Clément Cancès, Claire​​ Chainais-Hillairet, Amélie Dupouy​​​‌, Maxime Herda,‌ Juliette Venel.

In‌​‌ 16, Maxime Herda​​ , Claire Chainais-Hillairet  et​​​‌ al. present the numerical‌ analysis and simulations of‌​‌ a multi-dimensional memristive device​​ model. Memristive devices and​​​‌ memtransistors based on two-dimensional‌ (2D) materials have demonstrated‌​‌ promising potential for neuromorphic​​ computing and nextgeneration memory​​​‌ technologies. Their charge transport‌ model describes the drift-diffusion‌​‌ of electrons, holes, and​​ ionic defects self-consistently in​​​‌ an electric field. They‌ incorporate two types of‌​‌ boundary models: ohmic and​​ Schottky contacts. The coupled​​​‌ drift-diffusion partial differential equations‌ are discretized using a‌​‌ physics-preserving Voronoi finite volume​​ method. It relies on​​​‌ an implicit time-stepping scheme‌ and the excess chemical‌​‌ potential flux approximation. They​​ demonstrate that the fully​​​‌ discrete nonlinear scheme is‌ unconditionally stable, preserving the‌​‌ free-energy structure of the​​ continuous system and ensuring​​​‌ the nonnegativity of carrier‌ densities. Novel discrete entropy-dissipation‌​‌ inequalities for both boundary​​ condition types in multiple​​​‌ dimensions allow us to‌ prove the existence of‌​‌ discrete solutions. They perform​​ multi-dimensional simulations to understand​​​‌ the impact of electrode‌ configurations and device geometries,‌​‌ focusing on the hysteresis​​ behavior in lateral 2D​​​‌ memristive devices. Three electrode‌ configurations – side, top,‌​‌ and mixed contacts –​​ are compared numerically for​​​‌ different geometries and boundary‌ conditions. These simulations reveal‌​‌ the conditions under which​​ a simplified one-dimensional electrode​​​‌ geometry can well represent‌ the three electrode configurations.‌​‌ This work lays the​​ foundations for developing accurate,​​​‌ efficient simulation tools for‌ 2D memristive devices and‌​‌ memtransistors, offering tools and​​ guidelines for their design​​​‌ and optimization in future‌ applications.

In 41,‌​‌ Maxime Herda  et al.​​ design, analyze and simulate​​​‌ a finite volume scheme‌ for a cross-diffusion system‌​‌ which models chemotaxis with​​ local sensing. This system​​​‌ has the same Lyapunov‌ function (or entropy) as‌​‌ the celebrated minimal Keller-Segel​​ system, but unlike the​​​‌ latter, its solutions are‌ known to exist globally‌​‌ in 2D. The long-time​​ behavior of solutions is​​​‌ only partially understood which‌ motivates numerical exploration with‌​‌ a reliable numerical method.​​ They propose a linearly​​​‌ implicit, two-point flux finite‌ volume approximation of the‌​‌ system. They show that​​ the scheme preserves, at​​​‌ the discrete level, the‌ main features of the‌​‌ continuous system, namely mass​​ conservation, non-negativity of solution,​​​‌ entropy dissipation, and duality‌ estimates. These properties allow‌​‌ them to prove the​​ wellposedness, unconditional stability and​​​‌ convergence of the scheme.‌ They also show rigorously‌​‌ that the scheme possesses​​ an asymptotic preserving (AP)​​​‌ property in the quasi-stationary‌ limit. They complement their‌​‌ analysis with thorough numerical​​​‌ experiments investigating convergence and​ AP properties of the​‌ scheme as well as​​ its reliability with respect​​​‌ to stability properties of​ steady solutions.

In 48​‌Alain Blaustein , Claire​​ Chainais-Hillairet , Maxime Herda​​​‌  et al. consider a​ stationary drift-diffusion system with​‌ ionic charge carriers and​​ external generation of electron​​​‌ and hole charge carriers.​ This system arises, among​‌ other applications, in the​​ context of semiconductor modeling​​​‌ for perovskite solar cells.​ Thanks to truncation techniques​‌ and iterative energy estimates,​​ they show the existence​​​‌ and uniform upper and​ lower bounds on the​‌ solutions. The dependency of​​ the bounds on the​​​‌ various parameters of the​ model is investigated numerically​‌ on physically relevant test​​ cases.

In 64,​​​‌ Juliette Venel  et al.​ initiate a mathematical investigation​‌ of a PDE model​​ for the transport of​​​‌ high voltage direct current​ via overhead lines. They​‌ prove the existence of​​ infinitely many solutions, give​​​‌ necessary conditions for existence,​ explicitly compute the continuum​‌ of all radial solutions,​​ and develop a new​​​‌ numerical algorithm for this​ problem.

In 53,​‌ Clément Cances , Claire​​ Chainais-Hillairet , and Amélie​​​‌ Dupouy study a toy​ model for the evolution​‌ of the oxygen concentration​​ in an oxide layer.​​​‌ It consists in a​ transient convection diffusion equation​‌ in a one-dimensional domain​​ of variable width. The​​​‌ motions of the boundaries​ are governed by the​‌ traces of the concentration.​​ The authors exhibit a​​​‌ necessary and sufficient condition​ on the parameters involved​‌ in the model for​​ the existence of a​​​‌ unique traveling-wave solution. Moreover,​ they show that the​‌ model admits some universal​​ entropy structure, in the​​​‌ sense that any convex​ function of the concentration​‌ yields a dissipated free​​ energy (up to exchanges​​​‌ with the outer environment​ at the boundaries). Then​‌ the authors propose an​​ implicit in time arbitrary​​​‌ Lagrangian-Eulerian finite volume scheme​ based on Scharfetter-Gummel fluxes.​‌ It is shown to​​ be unconditionally convergent, to​​​‌ preserve exactly the travelling​ wave, and to dissipate​‌ all the aforementioned free​​ energies. Numerical experiments show​​​‌ that the scheme is​ first order accurate in​‌ time and second order​​ in space, and that​​​‌ the transient solution converges​ in the long-time limit​‌ towards the traveling-wave solution.​​

The physical processes of​​​‌ heat conduction, liquid water​ percolation, and phase changes​‌ govern the transfer of​​ mass and energy in​​​‌ snow. They are therefore​ at the heart of​‌ any physics-based snowpack model.​​ In the last decade,​​​‌ the use of Richards'​ equation has been proposed​‌ to better represent liquid​​ water percolation in snow.​​​‌ While this approach allows​ the explicit representation of​‌ capillary effects, it can​​ also be problematic as​​​‌ it usually presents a​ large increase in numerical​‌ complexity and cost. This​​ notably arises from the​​​‌ problem of applying a​ water retention curve in​‌ a fully-dry medium such​​ as snow, leading to​​​‌ a divergence and degeneracy​ in Richards' equation. Moreover,​‌ the difficulty of representing​​ both dry and wet​​​‌ snow in a single​ framework hinders the concomitant​‌ solving of heat conduction,​​ phase changes, and liquid​​ percolation. Rather, current models​​​‌ employ a sequential approach,‌ which can be subject‌​‌ to non-physical overshoots. To​​ treat these problems, Clément​​​‌ Cancès  et al. propose‌ in 66 the use‌​‌ of a regularized water​​ retention curve (WRC), that​​​‌ can be applied to‌ dry snow. Combined with‌​‌ a variable switch technique,​​ this opens the possibility​​​‌ of solving the energy‌ and mass budgets in‌​‌ a fully consistent and​​ tightly coupled manner, whether​​​‌ the snowpack contains dry‌ and/or wet regions. To‌​‌ assess the behavior of​​ the proposed scheme, the​​​‌ authors compare it to‌ other implementations based on‌​‌ loose-coupling between processes and​​ on the state-of-the-art strategies​​​‌ in snowpack models. Results‌ show that the use‌​‌ of a regularized WRC​​ with a variable switch​​​‌ greatly improve the robustness‌ of the numerical implementation,‌​‌ consistently allowing for large​​ timesteps, which results in​​​‌ faster and cheaper simulations.‌

Entropy and the second‌​‌ principle of thermodynamics are​​ regularly used as an​​​‌ analysis tool in applied‌ mathematics for physics-based numerical‌​‌ models. In essence, this​​ approach states that the​​​‌ second principle (i.e. the‌ non-destruction of entropy) is‌​‌ closely related to stability.​​ Consequently, numerical models complying​​​‌ with the second principle‌ are expected to be‌​‌ more robust than models​​ that do not. A​​​‌ notable advantage of this‌ method is its straight-forward‌​‌ generalization to nonlinear physics​​ and to systems of​​​‌ coupled equations. The goal‌ of the work 67‌​‌ by Clément Cancès  et​​ al. is to thus​​​‌ investigate the added-value of‌ such an entropybased analysis‌​‌ to the case of​​ snowpack modelling. For that,​​​‌ the authors study the‌ conditions under which the‌​‌ physics describing snowpacks respects​​ the second principle and​​​‌ the numerical schemes that‌ preserve this compliance after‌​‌ temporal and spatial discretization.​​ Specifically, we consider three​​​‌ cases of increasing complexity:‌ (i) a dry snowpack‌​‌ governed by heat conduction​​ only (meant to be​​​‌ an example of the‌ method for unfamiliar readers),‌​‌ (ii) a system composed​​ of a canopy and​​​‌ a snowpack exchanging heat,‌ and (iii) a dry‌​‌ snowpack with heat conduction,​​ vapor diffusion, and ice-vapor​​​‌ phase changes.

Multiphase chemical‌ equilibrium problems lead to‌​‌ nonlinear systems with complementarity​​ constraints, which become particularly​​​‌ challenging when phases may‌ vanish. In 72,‌​‌ Clément Cancès  et al.​​ introduce a new algebraic​​​‌ formulation of the equilibrium‌ problem based on extended‌​‌ mole fractions, derived from​​ the subdifferential of the​​​‌ Gibbs free energy, and‌ establish its equivalence with‌​‌ the classical minimization problem.​​ Their analysis provides new​​​‌ conditions ensuring the uniqueness‌ of solutions, even when‌​‌ some phases disappear. Building​​ on this formulation, the​​​‌ authors propose two parametrized‌ Newton-based strategies: one reformulates‌​‌ the relation between species​​ quantities and chemical potentials,​​​‌ while the other parametrizes‌ the complementarity conditions directly.‌​‌ Numerical experiments on a​​ system with 72 species​​​‌ and 22 phases confirm‌ the robustness and efficiency‌​‌ of the proposed methods.​​ In tests with randomized​​​‌ inputs, both strategies achieve‌ success rates above 90%‌​‌ with moderate iteration counts,​​ outperforming established approaches such​​​‌ as the Newton-min and‌ Fischer-Burmeister complementarity functions, and‌​‌ interior-point methods.

7.2 Analysis​​​‌ and numerical simulation in​ electromagnetism and related fields​‌

Participants: Théophile Chaumont-Frelet,​​ Marien-Lorenzo Hanot, Simon​​​‌ Lemaire.

In 25​Théophile Chaumont-Frelet proposes a​‌ novel a posteriori error​​ estimator for the Nédélec​​​‌ finite element discretization of​ time-harmonic Maxwell's equations. After​‌ the approximation of the​​ electric field is computed,​​​‌ he proposes a fully​ localized algorithm to reconstruct​‌ approximations to the electric​​ displacement and the magnetic​​​‌ field, with such approximations​ respectively fulfilling suitable divergence​‌ and curl constraints. These​​ reconstructed fields are in​​​‌ turn used to construct​ an a posteriori error​‌ estimator which is shown​​ to be reliable and​​​‌ efficient. Specifically, the estimator​ controls the error from​‌ above up to a​​ constant that tends to​​​‌ one as the mesh​ is refined and/or the​‌ polynomial degree is increased,​​ and from below up​​​‌ to constant independent of​ $p$. Both bounds​‌ are also fully-robust in​​ the low-frequency regime. The​​​‌ properties of the proposed​ estimator are illustrated on​‌ a set of numerical​​ examples.

In 56,​​​‌ Théophile Chaumont-Frelet , Jérôme​ Droniou and Simon Lemaire​‌ establish Maxwell compactness results​​ for the Discrete De​​​‌ Rham (DDR) polytopal complex:​ sequences in this polytopal​‌ complex with bounded discrete​​ $H\left(c\mathrm{u‌}rl\right)$ (resp.​ discrete $H(\mathrm{c‌}url)$) norm and orthogonal​​​‌ to discrete gradients (resp.​ discrete curls) have ${\mathrm{L‌}}^2$-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. The​​​‌ 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.​

The CHDG method is​‌ a hybridizable discontinuous Galerkin​​ (HDG) finite element method​​​‌ suitable for the iterative​ solution of time-harmonic wave​‌ propagation problems. Hybrid unknowns​​ corresponding to transmission variables​​​‌ are introduced at the​ element interfaces and the​‌ physical unknowns inside the​​ elements are eliminated, resulting​​​‌ in a hybridized system​ with favorable properties for​‌ fast iterative solution. In​​ 75, Théophile Chaumont-Frelet​​​‌  et al. extend the​ CHDG method, initially studied​‌ for the Helmholtz equation,​​ to the time-harmonic Maxwell​​​‌ equations. They prove that​ the local problems stemming​‌ from hybridization are well-posed​​ and that the fixed-point​​​‌ iteration naturally associated to​ the hybridized system is​‌ contractive. They propose a​​ 3D implementation with a​​​‌ discrete scheme based on​ nodal basis functions. The​‌ resulting solver and different​​ iterative strategies are studied​​​‌ with several numerical examples​ using a high-performance parallel​‌ C++ code.

In 44​​, Simon Lemaire and​​​‌ Silvano Pitassi prove discrete​ versions of the first​‌ and second Weber inequalities​​ on $H(\mathrm{c}url)‌\cap H(\mathrm{d‌}iv)$-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 $\mathrm{H}\left(cu\mathrm{r‌}l\right)$- and‌ $H\left(d\mathrm{i‌‌}v\right)$-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. The results‌​‌ are valid on domains​​ with general, possibly non-trivial​​​‌ topology. In a second‌ part, related discrete Maxwell‌​‌ compactness results are also​​ proved, still within a​​​‌ general topological setting.

In‌ 35, Simon Lemaire‌​‌ , Silvano Pitassi et​​ al. devise and analyze​​​‌ hybrid polyhedral methods of‌ arbitrary order for the‌​‌ approximation of div-curl systems​​ on three-dimensional domains with​​​‌ non-trivial topology. The div-curl‌ systems under consideration stem‌​‌ from magnetostatics, and can​​ either be first-order (for​​​‌ field formulations) or second-order‌ (for vector potential formulations).‌​‌ The mathematical analysis crucially​​ leverages the discrete Weber​​​‌ inequalities established in 44‌. An in-depth numerical‌​‌ assessment of the approach​​ is also performed, covering,​​​‌ in particular, the tricky‌ case of non-simply-connected domains.‌​‌

In  65, Marien-Lorenzo​​ Hanot  et al. prove​​​‌ discrete Poincaré inequalities that‌ are uniform in mesh‌​‌ size for the discrete​​ de Rham complex of​​​‌ differential forms, extending known‌ inequalities for gradient, curl,‌​‌ and divergence operators to​​ polytopal domains of arbitrary​​​‌ dimension and topology. A‌ crucial part of the‌​‌ proof involves deriving Poincaré​​ inequalities for the cochain​​​‌ complex supported on the‌ polytopal mesh. These inequalities‌​‌ have independent significance, aiding​​ in the existence and​​​‌ stability analysis of solutions‌ to numerical schemes such‌​‌ as Mimetic Finite Differences,​​ Compatible Discrete Operators, and​​​‌ Discrete Geometric Approaches.

In‌  70, Marien-Lorenzo Hanot‌​‌  et al. provide a​​ reformulation of the linearized​​​‌ Arnowitt–Deser–Misner equations as a‌ Hodge–Dirac wave system built‌​‌ upon the divdiv complex,​​ a structure that naturally​​​‌ handles issues of gauge‌ fixing, constraint propagation, and‌​‌ tensor symmetries in numerical​​ relativity. The well-posedness of​​​‌ the resulting system is‌ established, and a discretization‌​‌ strategy with error estimates​​ under minimal assumptions is​​​‌ proposed.

7.3 Analysis of‌ dissipative models and their‌​‌ discretization

Participants: Matthieu Alfaro​​, Alain Blaustein,​​​‌ Maxime Herda, Andrea‌ Natale, Marc Pegon‌​‌, Juliette Venel.​​

In 71, Maxime​​​‌ Herda , Marc Pegon‌  et al. provide a‌​‌ result of exponential stability​​ for several dissipative linear​​​‌ kinetic equations with heavy-tailed‌ equilibria. The approach, inspired‌​‌ by the so-called ${\mathrm{L}}^2$-hypocoercivity method, is​​​‌ robust enough to provide‌ estimates that are uniform‌​‌ in the anomalous diffusion​​ limit. Moreover, it is​​​‌ able to deal with‌ bounded domains with periodic‌​‌ boundary condition or general​​ Maxwell boundary condition (from​​​‌ the pure specular to‌ the pure diffusive case).‌​‌ In addition, this framework​​ accommodates linear collisional operators​​​‌ that act simultaneously on‌ the velocity and spatial‌​‌ variables.

In 18,​​​‌ Matthieu Alfaro , Maxime​ Herda , and Andrea​‌ Natale consider an epidemic​​ model with distributed-contacts. When​​​‌ the contact kernel concentrates,​ one formally reaches a​‌ very degenerate Fisher-KPP equation​​ with a diffusion term​​​‌ that is not in​ divergence form. They make​‌ an exhaustive study of​​ its travelling waves. For​​​‌ every admissible speed, there​ exist not only a​‌ unique non-saturated (smooth) wave​​ but also infinitely many​​​‌ saturated (sharp) ones. Furthermore​ their tails may differ​‌ from what is usually​​ expected. These results are​​​‌ thus in sharp contrast​ with their counterparts on​‌ related models.

In 50​​, Alain Blaustein  et​​​‌ al. design an asymptotic​ preserving scheme for the​‌ Vlasov-Poisson system in the​​ quasineutral regime. They first​​​‌ establish a convergence result​ for the continuous solution​‌ with optimal error estimates.​​ Following this path, they​​​‌ propose a fully discrete​ numerical method and rigorously​‌ prove that it is​​ uniformly consistent. Finally, they​​​‌ perform several numerical simulations​ to illustrate the behavior​‌ of the proposed scheme​​ which confirm the theoretical​​​‌ findings: stability and asymptotic​ preservation.

In  74,​‌ Juliette Venel  et al.​​ study the the long-term​​​‌ safety of the geological​ repository of nuclear wastes.​‌ A diffusion equation with​​ a moving free boundary​​​‌ in one dimension is​ introduced and studied. The​‌ model describes some mechanisms​​ involved in corrosion processes​​​‌ at the surface of​ carbon steel canisters in​‌ contact with a claystone​​ formation. The main objective​​​‌ of the paper is​ to prove the existence​‌ of weak solutions to​​ the problem which are​​​‌ maximal in time. For​ this, a time semidiscrete​‌ minimizing movements scheme based​​ on a Wasserstein-like distance​​​‌ is introduced. The existence​ of solutions to the​‌ scheme is proved. Then,​​ using a priori estimates,​​​‌ it is shown that​ as the time step​‌ goes to zero these​​ solutions converge up to​​​‌ extraction towards a maximal​ weak solution to the​‌ free boundary model.

7.4​​ A posteriori error analysis​​​‌

Participants: Théophile Chaumont-Frelet.​

In 24Théophile Chaumont-frelet​‌ proposes new a posteriori​​ error estimators for non-conforming​​​‌ finite element discretizations of​ second-order elliptic PDE problems.​‌ These estimators are based​​ on novel reformulations of​​​‌ the standard Prager-Synge identity,​ and enable to prove​‌ efficiency estimates without extra​​ stabilization terms in the​​​‌ error measure for a​ large class of discretization​‌ schemes. He proposes a​​ residual-based estimator for which​​​‌ the efficiency constant scales​ optimally in polynomial degree,​‌ as well as two​​ equilibrated estimators that are​​​‌ polynomial-degree-robust. One of the​ two estimators further leads​‌ to guaranteed error bounds.​​

In 55, Théophile​​​‌ Chaumont-Frelet considers second-order PDE​ problems set in unbounded​‌ domains and discretized by​​ Lagrange finite elements on​​​‌ a finite mesh, thus​ introducing an artificial boundary​‌ in the discretization. Specifically,​​ he considers the reaction​​​‌ diffusion equation as well​ as Helmholtz problems in​‌ waveguides with perfectly matched​​ layers. The usual procedure​​​‌ to deal with such​ problems is to first​‌ consider a modeling error​​ due to the introduction​​​‌ of the artificial boundary,​ and estimate the remaining​‌ discretization error with a​​ standard a posteriori technique.​​ A shortcoming of this​​​‌ method, however, is that‌ it is typically hard‌​‌ to obtain sharp bounds​​ on the modeling error.​​​‌ In this work, he‌ proposes a new technique‌​‌ that allows to control​​ the whole error by​​​‌ an a posteriori error‌ estimator. Specifically, he proposes‌​‌ a flux-equilibrated estimator that​​ is slightly modified to​​​‌ handle the truncation boundary.‌ For the reaction diffusion‌​‌ equation, we obtain fully-computable​​ guaranteed error bounds, and​​​‌ the estimator is locally‌ efficient and polynomial-degree-robust provided‌​‌ that the elements touching​​ the truncation boundary are​​​‌ not too refined. This‌ last condition may be‌​‌ seen as an extension​​ of the notion of​​​‌ shape-regularity of the mesh,‌ and does not prevent‌​‌ the design of efficient​​ adaptive algorithms. For the​​​‌ Helmholtz problem, as usual,‌ these statements remain valid‌​‌ if the mesh is​​ sufficiently refined. The theoretical​​​‌ findings are completed with‌ numerical examples which indicate‌​‌ that the estimator is​​ suited to drive optimal​​​‌ adaptive mesh refinements.

In‌ 57, Théophile Chaumont-Frelet‌​‌  et al. analyse adaptive​​ combined field integral equations​​​‌ for Helmholtz problems. While‌ the exterior Helmholtz problem‌​‌ with Dirichlet boundary conditions​​ is always well-posed, the​​​‌ associated standard boundary integral‌ equations are not if‌​‌ the squared wavenumber agrees​​ with an eigenvalue of​​​‌ the interior Dirichlet problem.‌ Combined field integral equations‌​‌ are not affected by​​ this spurious resonances but​​​‌ are essentially restricted to‌ sufficiently smooth boundaries. For‌​‌ general Lipschitz domains, the​​ latter integral equations are​​​‌ applicable through suitable regularization.‌ Under fairly general assumptions‌​‌ on the regularizing operator,​​ they propose a posterirori​​​‌ computable error estimators for‌ corresponding Galerkin boundary element‌​‌ methods of arbitrary polynomial​​ degree. We show that​​​‌ adaptive mesh-refining algorithms steered‌ by these local estimators‌​‌ converge at optimal algebraic​​ rate with respect to​​​‌ the number of underlying‌ boundary mesh elements. In‌​‌ particular, they consider mixed​​ formulations involving the inverse​​​‌ Laplace-Beltrami as regularizing operator.‌ Numerical examples highlight that‌​‌ in the vicinity of​​ spurious resonances the proposed​​​‌ adaptive algorithm is significantly‌ more performant when applied‌​‌ to the regularized combined​​ field equation rather than​​​‌ the standard one.

In‌ 58, Théophile Chaumont-Frelet‌​‌  et al. consider linear​​ reaction-diffusion equations posed on​​​‌ unbounded domains, and discretized‌ by adaptive Lagrange finite‌​‌ elements. To obtain finite-dimensional​​ spaces, it is necessary​​​‌ to introduce a truncation‌ boundary, whereby only a‌​‌ bounded computational subdomain is​​ meshed, leading to an​​​‌ approximation of the solution‌ by zero in the‌​‌ remainder of the domain.​​ They propose a residual-based​​​‌ error estimator that accounts‌ for both the standard‌​‌ discretization error as well​​ as the effect of​​​‌ the truncation boundary. This‌ estimator is shown to‌​‌ be reliable and efficient​​ under appropriate assumptions on​​​‌ the triangulation. Based on‌ this estimator, they devise‌​‌ an adaptive algorithm that​​ automatically refines the mesh​​​‌ and pushes the truncation‌ boundary towards infinity. They‌​‌ prove that this algorithm​​ converges and even achieves​​​‌ optimal rates in terms‌ of the number of‌​‌ degrees of freedom. They​​ finally provide numerical examples​​​‌ illustrating the key theoretical‌ findings.

In 59,‌​‌ Théophile Chaumont-Frelet  et al.​​​‌ consider a two dimensional​ biharmonic problem and its​‌ discretization by means of​​ a symmetric interior penalty​​​‌ discontinuous Galerkin method. Based​ on the "div-div" complex,​‌ a novel split of​​ an error measure based​​​‌ on a generalized Hessian​ into two terms measuring​‌ the conformity and nonconformity​​ of the scheme is​​​‌ proven. This splitting is​ the departing point for​‌ the design of a​​ new reliable and efficient​​​‌ error estimator, which does​ not involve any stabilization.​‌ Such an error estimator​​ can be bounded from​​​‌ above by the standard​ residual error estimator. Numerical​‌ results assess the theoretical​​ predictions, including the efficiency​​​‌ of the proposed estimator.​

In 60, Théophile​‌ Chaumont-Frelet  et al. study​​ a posteriori error estimates​​​‌ and their use for​ time-dependent acoustic scattering problems,​‌ formulated as a time-dependent​​ boundary integral equation based​​​‌ on a single-layer ansatz.​ The integral equation is​‌ discretized by the convolution​​ quadrature method in time​​​‌ and by boundary elements​ in space. They prove​‌ the reliability of an​​ error estimator of residual​​​‌ type and study the​ resulting space-adaptive mesh refinements.​‌ Moreover, they present a​​ simple modification of the​​​‌ convolution quadrature method based​ on temporal shifts, which​‌ recovers, for the boundary​​ densities, the full classical​​​‌ temporal convergence order $\mathrm{2}m-1$ of​‌ the temporal convolution quadrature​​ method based on the​​​‌ $m$-stage convolution quadrature​ semi-discretization. They numerically observe​‌ that the adaptive scheme​​ yields asymptotically optimal meshes​​​‌ for an acoustic scattering​ problem in two dimensions.​‌

7.5 Analysis and numerical​​ simulation of variational models​​​‌

Participants: Marc Pegon.​

In 52, Marc​‌ Pegon  et al. investigate​​ generalized liquid drop models​​​‌ with screened Riesz-type interactions,​ focusing in particular on​‌ truncated Coulomb and Yukawa​​ potentials in three dimensions.​​​‌ They establish the existence​ of non spherical minimizers​‌ for some values of​​ the screening parameter. This​​​‌ gives the first evidence​ of such minimizers in​‌ the class of repulsive,​​ radial, and radially nonincreasing​​​‌ kernels in three dimensions.​ They further show that​‌ in the classical Riesz​​ case, the widely-believed conjecture​​​‌ that minimizers are either​ balls or do not​‌ exist remains consistent with​​ their results, but only​​​‌ just. Indeed they observe​ that the energy-per-mass ratios​‌ of the best balls​​ and of the best​​​‌ cylinders are surprisingly close.​

7.6 Complementary topics in​‌ numerical and PDE analysis​​

Participants: Matthieu Alfaro,​​​‌ Claire Chainais-Hillairet, Cindy​ Guichard, Théophile Chaumont-Frelet​‌, Emmanuel Creusé,​​ Andrea Natale, Juliette​​​‌ Venel.

In 49​, Matthieu Alfaro ,​‌ Claire Chainais-Hillairetet. al​​ consider the so-called field-road​​​‌ diffusion model in a​ bounded domain, consisting of​‌ two parabolic PDEs posed​​ on sets of different​​​‌ dimensions and coupled through​ (symmetric) nonlinear exchange terms.​‌ They propose a new​​ and rather direct functional​​​‌ inequalities approach to prove​ the exponential decay of​‌ a relative entropy, and​​ thus the convergence of​​​‌ the solution towards the​ stationary state selected by​‌ the total mass of​​ the initial datum.

In​​​‌ 30, Théophile Chaumont-Frelet​et al. investigate the​‌ approximation properties of solutions​​ to the Ginzburg-Landau equation​​ (GLE) in finite element​​​‌ spaces. Special attention is‌ given to how the‌​‌ errors are influenced by​​ coupling the mes$\mathrm{h‌}$ size h and the‌ polynomial degree $p$ of‌​‌ the finite element space​​ to the size of​​​‌ the so-called Ginzburg-Landau material‌ parameter $\kappa$. As‌​‌ observed in previous works,​​ the finite element approximations​​​‌ to the GLE are‌ suffering from a numerical‌​‌ pollution effect, that is,​​ the best-approximation error in​​​‌ the finite element space‌ converges under mild coupling‌​‌ conditions between $h$ and​​ $\kappa$, whereas the​​​‌ actual finite element solutions‌ possess poor accuracy in‌​‌ a large pre-asymptotic regime​​ which depends on $\mathrm{\kappa ‌}$. In this paper,‌ they provide a new‌​‌ error analysis that allows​​ them to quantify the​​​‌ preasymptotic regime and the‌ corresponding pollution effect in‌​‌ terms of explicit resolution​​ conditions. In particular, they​​​‌ are able to prove‌ that higher polynomial degrees‌​‌ reduce the pollution effect,​​ i.e., the accuracy of​​​‌ the best-approximation is achieved‌ under relaxed conditions for‌​‌ the mesh size. They​​ provide both error estimates​​​‌ in the $H^{\mathrm{1‌}}$- and the ${\mathrm{L‌‌}}^2$-norm and illustrate​​ their findings with numerical​​​‌ examples.

In 32,‌ Théophile Chaumont-Frelet  et al.‌​‌ design a quasi-interpolation operator​​ from the Sobolev space​​​‌ $H_0^1$ to‌ its finite-dimensional finite element‌​‌ subspace formed by piecewise​​ polynomials on a simplicial​​​‌ mesh with a computable‌ approximation constant. The operator‌​‌ 1) is defined on​​ the entire $H_{\mathrm{0‌}}^1$, no additional‌ regularity is needed; 2)‌​‌ allows for an arbitrary​​ polynomial degree; 3) works​​​‌ in any space dimension;‌ 4) is defined locally,‌​‌ in vertex patches of​​ mesh elements; 5) yields​​​‌ optimal estimates for both‌ the $H^1$ seminorm‌​‌ and the $L^{\mathrm{2}}$ norm error; 6) gives​​​‌ a computable constant for‌ both the $H^{\mathrm{1‌‌}}$ seminorm and the ${\mathrm{L}}^2$ norm error; 7)​​​‌ leads to the equivalence‌ of global-best and local-best‌​‌ errors; 8) possesses the​​ projection property. Its construction​​​‌ follows the so-called potential‌ reconstruction from a posteriori‌​‌ error analysis. Numerical experiments​​ illustrate that our quasi-interpolation​​​‌ operator systematically gives the‌ correct convergence rates in‌​‌ both the $H^{\mathrm{1}}$ seminorm and the ${\mathrm{L‌}}^2$ norm and its‌ certified overestimation factor is‌​‌ rather sharp and stable​​ in all tested situations.​​​‌

In 54, Théophile‌ Chaumont-Frelet analysis the well-posedness‌​‌ (or lack thereof) of​​ three-dimensional time-harmonic wave propagation​​​‌ problems modeled by the‌ Helmholtz equation. It is‌​‌ well-known that if the​​ problem is set in​​​‌ bounded domain with Dirichlet‌ boundary conditions, then the‌​‌ Helmholtz problem is well-posed​​ for all (real-valued) frequencies​​​‌ except for a sequence‌ of countably many resonant‌​‌ frequencies that accumulate at​​ infinity. In fact, if​​​‌ the domain is sufficiently‌ smooth, this can be‌​‌ quantified further and Weyl's​​ law states that the​​​‌ number of resonant frequencies‌ less than a given‌​‌ $k>0$ scales​​ as $k^3$.​​​‌ On the other hand,‌ scattering problems set in‌​‌ ${ℝ}^3$ with a​​ radiation condition at infinity​​​‌ and a bounded obstacle‌ modeled by variations in‌​‌ the PDE coefficients are​​​‌ well-posed for all frequencies​ under mild regularity assumption​‌ on such coefficients. In​​ 2001, Filinov provided a​​​‌ counter example of a​ rough coefficient such that​‌ the scattering problem is​​ not well-posed for (at​​​‌ least) a single frequency​ $k$. In this​‌ contribution, he uses this​​ result to show that​​​‌ for all $\epsilon >0$ one can design​‌ a rough coefficient corresponding​​ to a compactly supported​​​‌ obstacle such that the​ scattering problem is ill-posed​‌ for a countable sequence​​ of frequencies accumulating at​​​‌ infinity, and such that​ the number of such​‌ frequencies less than any​​ given $k>\mathrm{0‌}$ scales as $k^{\mathrm{3}-\epsilon }$.

In​‌ 62, Théophile Chaumont-Frelet​​et. al. construct potentials​​​‌ for the exterior derivative,​ in particular, for the​‌ gradient, the curl, and​​ the divergence operators, over​​​‌ domains with shellable triangulations.​ Notably, the class of​‌ shellable triangulations includes local​​ patches (stars) in two​​​‌ or three dimensions. The​ operator norms of the​‌ potentials satisfy explicitly computable​​ bounds that depend only​​​‌ on the geometry. They​ thus compute upper bounds​‌ for constants in Poincaré-Friedrichs​​ inequalities and lower bounds​​​‌ for the eigenvalues of​ vector Laplacians. As an​‌ additional result with independent​​ standing, they establish Poincaré-Friedrichs​​​‌ inequalities with computable constants​ for the $L^{\mathrm{p‌}}$ de Rham complex over​​ bounded convex domains, derived​​​‌ as explicit operator norms​ of regularized Poincaré and​‌ Bogovskii potential operators. They​​ express all our main​​​‌ results in the calculus​ of differential forms and​‌ treat the gradient, curl,​​ and divergence operators as​​​‌ instances of the exterior​ derivative. Computational examples illustrate​‌ the theoretical findings.

In​​ 63, Théophile Chaumont-Frelet​​​‌  et al. introduce a​ quasi-interpolation operator that maps​‌ the infinite-dimensional Sobolev space​​ $H^1$ into its​​​‌ finite-dimensional Lagrange finite element​ subspace formed by piecewise​‌ polynomials on a tetrahedral​​ mesh. This operator enjoys​​​‌ the following key properties:​ 1) it is defined​‌ over the entire ${\mathrm{H}}^1$ and includes essentially​​​‌ boundary conditions imposed on​ a part of the​‌ boundary; 2) it is​​ defined locally in a​​​‌ neighborhood of each tetrahedron​ of the mesh; 3)​‌ it is based on​​ simple piecewise polynomial projections;​​​‌ 4) it is stable​ in the $H^{\mathrm{1‌}}$-seminorm; 5) it has​​ optimal (local-best) approximation properties;​​​‌ 6) it commutes with​ its sibling operator on​‌ $H\left(c\mathrm{u}rl\right)$;​​​‌ 7) it is a​ projector.

In 61Théophile​‌ Chaumont-Frelet propose an algorithm​​ to numerically determined whether​​​‌ a second-order linear PDE​ problem satisfying a Gårding​‌ inequality is well-posed. This​​ algorithm further provides a​​​‌ lower bound to the​ inf-sup constant of the​‌ weak formulation, which may​​ in turn be used​​​‌ for a posteriori error​ estimation purposes. The numerical​‌ lower bound is based​​ on two discrete singular​​​‌ value problems involving a​ Lagrange finite element discretization​‌ coupled with an a​​ posteriori error estimator based​​​‌ on flux reconstruction techniques.​ He shows that if​‌ the finite element discretization​​ is sufficiently rich, the​​​‌ lower bound underestimates the​ optimal constant only by​‌ a factor roughly equal​​ to two.

In 33​​, Emmanuel Creusé ,​​​‌ Juliette Venel  et al.‌ consider the propagation of‌​‌ shock waves on nonuniform​​ grids. This phenomenon appears,​​​‌ for example, in some‌ impact problems of the‌​‌ fast-dynamics type, where a​​ finer mesh can typically​​​‌ be employed in regions‌ of interest, leading to‌​‌ mesh ratios that can​​ range from 10 to​​​‌ 20 or even more.‌ Unfortunately, unwanted spurious reflections‌​‌ occur for this type​​ of problem using (standard)​​​‌ explicit finite element software.‌ An optimized explicit Runge-Kutta-Nyström‌​‌ time-integration scheme is therefore​​ built to minimize spurious​​​‌ wave reflections for shock‌ wave propagation on these‌​‌ nonuniform grids.

In 51​​Andrea Natale  et al.​​​‌ study the problem of‌ reconstructing a Laguerre tessellation‌​‌ from the volumes and​​ the barycenters of its​​​‌ cells. They show that‌ any vector of barycenters‌​‌ arising from a Laguerre​​ tessellation with prescribed cell​​​‌ volumes can be understood‌ as an exposed point‌​‌ of a convex set​​ defined by convex order​​​‌ relations. Leveraging this characterization,‌ they formulate a projection‌​‌ problem that can be​​ solved efficiently and yields​​​‌ an approximation of the‌ desired reconstruction. Importantly, the‌​‌ same method can also​​ construct a Laguerre tessellation​​​‌ that fits prescribed volumes‌ and barycenters in cases‌​‌ where these are not,​​ a priori, related to​​​‌ any Laguerre tessellation. This‌ strategy may find application‌​‌ in the analysis of​​ experimental data from polycrystalline​​​‌ samples.

In 73Andrea‌ Natale  et al. provide‌​‌ a precise characterization of​​ the link between backward/forward​​​‌ Wasserstein projections in convex‌ order and the recently‌​‌ introduced metric extrapolation problem.​​ They use such a​​​‌ link to derive new‌ quantitative stability estimates for‌​‌ both problems.

In 69​​Cindy Guichard provides a​​​‌ general framework for second-order‌ elliptic problems, which includes‌​‌ a variety of boundary​​ conditions. She shows how​​​‌ one can apply mass-lumped‌ mixed finite element to‌​‌ this problem, and she​​ provides sufficient conditions for​​​‌ the convergence of such‌ a method. In particular,‌​‌ she exhibits convergence results​​ assuming two different type​​​‌ of assumptions: on one‌ hand, she shows convergence‌​‌ properties following the standard​​ analysis of mixed finite​​​‌ elements. On the other‌ hand, she provides conditions‌​‌ on one of the​​ approximation space, which also​​​‌ lead to some convergence‌ properties. She then formulates‌​‌ Abstract Gradient Discretization method​​ (AGDM) based on these​​​‌ mass-lumped mixed finite elements,‌ enabling her to apply‌​‌ this type of discretization​​ to a variety of​​​‌ nonlinear problems. Finally, she‌ illustrates her results by‌​‌ two examples. The first​​ one is a second-order​​​‌ elliptic problem with homogeneous‌ Neumann boundary conditions, discretized‌​‌ by Raviart-Thomas finite elements.​​ She shows on this​​​‌ example that mass-lumping leads‌ to classical finite volume‌​‌ schemes. The second one​​ is inspired by the​​​‌ elliptic part of a‌ model of shallow water‌​‌ flows with dispersive terms.​​ She applies on this​​​‌ example generalized operators, and‌ she proves the convergence‌​‌ of the method used​​ in the literature.

7.7​​​‌ Habilitation (HDR) theses

Participants:‌ Juliette Venel.

On‌​‌ December 16, Juliette Venel​​ defended her Habilitation à​​​‌ Diriger des Recherches (HDR),‌ entitled "Theoretical and Numerical‌​‌ Contributions to the Study​​​‌ of Differential Inclusions and​ Partial Differential Equations" at​‌ the Polytechnic University of​​ Hauts-de-France.

8.1 Bilateral contracts with​‌ industry

Participant: Clément Cances​​.

Clément Cances heads​​​‌ the MATHSOUT project (1M​ euros, 2024–2029) together with​‌ I. Faille (IFPEN). This​​ project involves academic partners​​​‌ (Inria, UniCA, CNRS) as​ well as IFPEN and​‌ the French Geological survey​​ (BRGM). Ngoc Do Quyen​​​‌ Dang started her PhD​ at IFPEN in November​‌ 2024 in the framework​​ of the MATHSOUT project.​​​‌ The contract follows the​ lines of the bilateral​‌ agreement between Inria and​​ IFPEN.

8.2 Bilateral grants​​​‌ with industry

Participant: Simon​ Lemaire.

In 2023,​‌ the PRCE project HIPOTHEC​​ (HIgh-order PO​​​‌lyhedral meTHods​ for Eddy C​‌urrent testing simulations) has​​ been funded in the​​​‌ generic ANR call. This​ 6-year project, which started​‌ in January 2024, is​​ coordinated by Simon Lemaire​​​‌ and is a collaboration​ between Inria, EDF, and​‌ 3 additional academic partners.​​ The aim of the​​​‌ project is to pursue,​ within an enlarged academic​‌ consortium, the research efforts​​ initiated in the framework​​​‌ of the France Relance​ collaboration between the team​‌ and EDF R&D. More​​ details about the HIPOTHEC​​​‌ project can be found​ in Section 9.3.2.​‌

9.1 International initiatives

9.2 International​‌ research visitors

9.3 National initiatives

Participants:‌​‌ Alain Blaustein, Clément​​ Cancès, Théophile Chaumont-Frelet​​​‌, Cindy Guichard,‌ Simon Lemaire, Andrea‌​‌ Natale, Marc Pegon​​.

10.1 Promoting​​​‌ scientific activities

10.2 Teaching - Supervision​‌ - Juries - Educational​​ and pedagogical outreach

10.3 Popularization

Alain​​ Blaustein hosted a high-school​​​‌ student at the Laboratoire​ Paul Painlevé for a​‌ half day during his​​ internship in October.

Claire​​​‌ Chainais-Hillairet is one of​ the organizers of the​‌ one-week internship "Les Fourmis​​ {éclairées}", which was held​​​‌ at Université de Lille​ in April 2025, details​‌ are available here.​​

Emmanuel Creusé co-organized the​​​‌ Sixth Valenciennes Mathematics Meetings.​ This event, bringing together​‌ mathematics students from the​​ UPHF and preparatory class​​​‌ students from the Wallon​ (Valenciennes) and Kastler (Denain)​‌ high schools, allows participants​​ to discover various aspects​​​‌ of mathematics through a​ series of scientific presentations​‌ given by teacher-researchers. The​​ program for the 2025​​​‌ edition is available here​.

Maxime Herda wrote​‌ popularization article on the​​ Inria ARISE research team​​​‌.

Simon Lemaire is​ currently (co-)Scientific Officer in​‌ charge of Mediation for​​ the Inria Lille research​​ center. He is responsible​​​‌ for both internal (organization‌ of the monthly scientific‌​‌ seminar "30 MIN. de​​ sciences") and external (through​​​‌ different events, intended in‌ particular for high school‌​‌ students) scientific outreach actions.​​ He co-organized in October​​​‌ 2025 the "Rendez-vous des‌ Jeunes Mathématiciennes et Informaticiennes"‌​‌ (RJMI). This​​ two-day event is specifically​​​‌ geared towards female high‌ school students, and aims‌​‌ at promoting scientific careers​​ amongst them. He also​​​‌ participated in the organization‌ of the Maths and‌​‌ Computer Science one-week internship​​ "Les Fourmis {éclairées}",​​​‌ which was held at‌ Université de Lille in‌​‌ April 2025. He is​​ finally invested in the​​​‌ national scientific outreach programme‌ "1 scientifique, 1 classe‌​‌ / Chiche !",​​ for which he realized​​​‌ an intervention in 2025.‌

In 2025, Marc Pegon‌​‌ co-animated two introductory research​​ activities for high school​​​‌ students on the Kakeya‌ needle problem.

On May‌​‌ 20, Juliette Venel presented​​ the various responsibilities of​​​‌ an associate professor during‌ the “Maths C for‌​‌ L” week, held at​​ the University of Artois​​​‌ in Lens.

11.1 Major publications‌​‌

• 1 articleM.Marianne​​ Bessemoulin-Chatard and C.Claire​​​‌ Chainais-Hillairet. Exponential decay‌ of a finite volume‌​‌ scheme to the thermal​​ equilibrium for drift--diffusion systems​​​‌.Journal of Numerical‌ Mathematics2532017‌​‌, 147-168HALDOI​​
• 2 articleM.Marianne​​​‌ Bessemoulin-Chatard, M.Maxime‌ Herda and T.Thomas‌​‌ Rey. Hypocoercivity and​​ diffusion limit of a​​​‌ finite volume scheme for‌ linear kinetic equations.‌​‌Mathematics of Computation89​​January 2020, 1093-1133​​​‌HALDOIback to‌ text
• 3 articleK.‌​‌Konstantin Brenner and C.​​Clément Cancès. Improving​​​‌ Newton's method performance by‌ parametrization: the case of‌​‌ Richards equation.SIAM​​ Journal on Numerical Analysis​​​‌5542017,‌ 1760--1785HALback to‌​‌ text
• 4 articleC.​​Clément Cancès, V.​​​‌Virginie Ehrlacher and L.‌Laurent Monasse. Finite‌​‌ Volumes for the Stefan-Maxwell​​ Cross-Diffusion System.IMA​​​‌ Journal of Numerical Analysis‌4422024,‌​‌ 1029–1060HALDOIback​​ to text
• 5 article​​​‌C.Clément Cancès,‌ T.Thomas Gallouët and‌​‌ L.Léonard Monsaingeon.​​ Incompressible immiscible multiphase flows​​​‌ in porous media: a‌ variational approach.Analysis‌​‌ & PDE108​​2017, 1845--1876HAL​​​‌DOIback to text‌
• 6 articleC.Clément‌​‌ Cancès and C.Cindy​​ Guichard. Numerical analysis​​​‌ of a robust free‌ energy diminishing Finite Volume‌​‌ scheme for parabolic equations​​ with gradient structure.​​​‌Foundations of Computational Mathematics‌1762017,‌​‌ 1525-1584HALback to​​ text
• 7 articleC.​​​‌Claire Chainais-Hillairet and M.‌Maxime Herda. Large-time‌​‌ behaviour of a family​​ of finite volume schemes​​​‌ for boundary-driven convection-diffusion equations‌.IMA Journal of‌​‌ Numerical Analysis404​​October 2020, 2473-2505​​​‌HALDOIback to‌ text
• 8 articleT.‌​‌Théophile Chaumont-Frelet and M.​​Martin Vohralík. p-robust​​​‌ equilibrated flux reconstruction in‌ H(curl) based on local‌​‌ minimizations. Application to a​​ posteriori analysis of the​​​‌ curl-curl problem.SIAM‌ Journal on Numerical Analysis‌​‌6142023,​​​‌ 1783--1818HALDOI
• 9​ articleF.Florent Chave​‌, D. A.Daniele​​ Antonio Di Pietro and​​​‌ S.Simon Lemaire.​ A discrete Weber inequality​‌ on three-dimensional hybrid spaces​​ with application to the​​​‌ HHO approximation of magnetostatics​.Mathematical Models and​‌ Methods in Applied Sciences​​3212022,​​​‌ 175-207HALDOI
• 10​ articleD. A.Daniele​‌ Antonio Di Pietro,​​ A.Alexandre Ern and​​​‌ S.Simon Lemaire.​ An arbitrary-order and compact-stencil​‌ discretization of diffusion on​​ general meshes based on​​​‌ local reconstruction operators.​Computational Methods in Applied​‌ Mathematics144June​​ 2014, 461-472HAL​​​‌DOIback to text​
• 11 articleG.Giacomo​‌ Dimarco, R.Raphaël​​ Loubère, J.Jacek​​​‌ Narski and T.Thomas​ Rey. An efficient​‌ numerical method for solving​​ the Boltzmann equation in​​​‌ multidimensions.Journal of​ Computational Physics3532018​‌, 46-81HALDOI​​
• 12 articleE.Enrico​​​‌ Facca, G.Gabriele​ Todeschi, A.Andrea​‌ Natale and M.Michele​​ Benzi. Efficient preconditioners​​​‌ for solving dynamical optimal​ transport via interior point​‌ methods.SIAM Journal​​ on Scientific Computing46​​​‌32024HALDOI​back to text
• 13​‌ articleT.Thomas Gallouët​​, A.Andrea Natale​​​‌ and G.Gabriele Todeschi​. From geodesic extrapolation​‌ to a variational BDF2​​ scheme for Wasserstein gradient​​​‌ flows.Mathematics of​ Computation932024,​‌ 2769-2810HALDOIback​​ to text
• 14 article​​​‌T.Thomas Gallouët,​ A.Andrea Natale and​‌ F.-X.François-Xavier Vialard.​​ Generalized compressible flows and​​​‌ solutions of the H(div)​ geodesic problem.Archive​‌ for Rational Mechanics and​​ Analysis2020HALDOI​​​‌
• 15 articleB.Benôit​ Merlet. A highly​‌ anisotropic nonlinear elasticity model​​ for vesicles I. Eulerian​​​‌ formulation, rigidity estimates and​ vanishing energy limit.​‌Arch. Ration. Mech. Anal.​​21722015,​​​‌ 651--680HALDOI

11.2​ Publications of the year​‌

11.3​​ Cited publications

• 76 article​​​‌A.Ahmed Ait Hammou‌ Oulhaj, C.Clément‌​‌ Cancès and C.Claire​​ Chainais-Hillairet. Numerical analysis​​​‌ of a nonlinearly stable‌ and positive Control Volume‌​‌ Finite Element scheme for​​ Richards equation with anisotropy​​​‌.ESAIM: Mathematical Modelling‌ and Numerical Analysis52‌​‌42018, 1532-1567​​HALDOIback to​​​‌ text
• 77 articleA.‌Ahmed Ait Hammou Oulhaj‌​‌ and D.David Maltese​​. Convergence of a​​​‌ positive nonlinear control volume‌ finite element scheme for‌​‌ an anisotropic seawater intrusion​​ model with sharp interfaces​​​‌.Numerical Methods for‌ Partial Differential Equations36‌​‌12020, 133-153​​HALDOIback to​​​‌ text
• 78 articleA.‌Ahmed Ait Hammou Oulhaj‌​‌. Numerical analysis of​​ a finite volume scheme​​​‌ for a seawater intrusion‌ model with cross-diffusion in‌​‌ an unconfined aquifer.​​Numerical Methods for Partial​​​‌ Differential EquationsMay 2018‌HALDOIback to‌​‌ text
• 79 bookL.​​Luigi Ambrosio, N.​​​‌Nicola Gigli and G.‌Giuseppe Savaré. Gradient‌​‌ flows: in metric spaces​​ and in the space​​​‌ of probability measures.‌Springer Science & Business‌​‌ Media2005back to​​ text
• 80 articleR.​​​‌R. Araya, C.‌C. Harder, D.‌​‌D. Paredes and F.​​F. Valentin. Multiscale​​​‌ Hybrid-Mixed method.SIAM‌ J. Numer. Anal.51‌​‌62013, 3505--3531​​back to text
• 81​​​‌ articleA.A. Arnold‌, P.P. Markowich‌​‌, G.G. Toscani​​ and A.A. Unterreiter​​​‌. On convex Sobolev‌ inequalities and the rate‌​‌ of convergence to equilibrium​​ for Fokker-Planck type equations​​​‌.Comm. Partial Differential‌ Equations261-22001‌​‌, 43-100URL: http://dx.doi.org/10.1081/PDE-100002246​​DOIback to text​​​‌
• 82 bookF.Franck‌ Assous, P.Patrick‌​‌ Ciarlet Jr. and S.​​Simon Labrunie. Mathematical​​​‌ foundations of computational electromagnetism‌.198Applied Mathematical‌​‌ SciencesSpringer, Cham2018​​, ix+458URL: https://doi.org/10.1007/978-3-319-70842-3​​​‌DOIback to text‌
• 83 articleN.Nathalie‌​‌ Ayi, M.Maxime​​ Herda, H.Hélène​​​‌ Hivert and I.Isabelle‌ Tristani. A note‌​‌ on hypocoercivity for kinetic​​ equations with heavy-tailed equilibrium​​​‌.Comptes Rendus. Mathématique‌3583July 2020‌​‌, 333-340HALDOI​​back to text
• 84​​​‌ articleN.Nathalie Ayi‌, M.Maxime Herda‌​‌, H.Hélène Hivert​​ and I.Isabelle Tristani​​​‌. On a structure-preserving‌ numerical method for fractional‌​‌ Fokker-Planck equations.Mathematics​​ of Computation92340​​​‌2023, 635--693HAL‌DOIback to text‌​‌
• 85 articleS.Sabrina​​ Bassetto, C.Clément​​​‌ Cancès, G.Guillaume‌ Enchéry and Q. H.‌​‌Quang Huy Tran.​​​‌ On several numerical strategies​ to solve Richards' equation​‌ in heterogeneous media with​​ Finite Volumes.Computational​​​‌ Geosciences262022,​ 1297--1322HALDOIback​‌ to text
• 86 inproceedings​​S.Sabrina Bassetto,​​​‌ C.Clément Cancès,​ G.Guillaume Enchéry and​‌ Q. H.Quang Huy​​ Tran. Robust Newton​​​‌ solver based on variable​ switch for a finite​‌ volume discretization of Richards​​ equation.Finite Volumes​​​‌ for Complex Applications IX​Bergen, NorwayJune 2020​‌HALback to text​​
• 87 articleC.C.​​​‌ Bataillon, F.F.​ Bouchon, C.C.​‌ Chainais-Hillairet, C.C.​​ Desgranges, E.E.​​​‌ Hoarau, F.F.​ Martin, S.S.​‌ Perrin, M.M.​​ Tupin and J.J.​​​‌ Talandier. Corrosion modelling​ of iron based alloy​‌ in nuclear waste repository​​.Electrochim. Acta55​​​‌152010, 4451--4467​back to textback​‌ to text
• 88 book​​J.J. Bear and​​​‌ Y.Y. Bachmat.​ Introduction to modeling of​‌ transport phenomena in porous​​ media.4Springer​​​‌1990back to text​
• 89 softwareL.Laurence​‌ Beaude and S.Simon​​ Lemaire. ParaSkel++: a​​​‌ C++ platform for the​ high-performance, arbitrary-order, 2/3D numerical​‌ approximation of PDEs on​​ general polytopal meshes using​​​‌ skeletal Galerkin methods.​v1August 2021 lic:​‌ GNU Lesser General Public​​ License v3.0 only.​​​‌HALSoftware HeritageVCS​back to text
• 90​‌ articleL.L. Beirão​​ da Veiga, F.​​​‌F. Brezzi, A.​A. Cangiani, G.​‌G. Manzini, L.​​ ..L .D. Marini​​​‌ and A.A. Russo​. Basic principles of​‌ virtual element methods.​​Math. Models Methods Appl.​​​‌ Sci. (M3AS)231​2013, 199--214back​‌ to text
• 91 incollection​​J.-D.Jean-David Benamou,​​​‌ G.Guillaume Carlier and​ F.Filippo Santambrogio.​‌ Variational mean field games​​.Active Particles, Volume​​​‌ 1Springer2017,​ 141--171back to text​‌
• 92 articleC.C.​​ Besse. A relaxation​​​‌ scheme for the nonlinear​ Schrödinger equation.42​‌32004, 934--952​​back to text
• 93​​​‌ articleM. C.Marianne​ Chatard Bessemoulin-Chatard and C.​‌Claire Chainais-Hillairet. Exponential​​ decay of a finite​​​‌ volume scheme to the​ thermal equilibrium for drift--diffusion​‌ systems.Journal of​​ Numerical Mathematics253​​​‌https://arxiv.org/abs/1601.00813September 2017,​ 147-168HALDOIback​‌ to text
• 94 article​​M.M. Bessemoulin-Chatard,​​​‌ C.C. Chainais-Hillairet and​ M.-H.M.-H. Vignal.​‌ Study of a finite​​ volume scheme for the​​​‌ drift-diffusion system. Asymptotic behavior​ in the quasi-neutral limit​‌.SIAM J. Numer.​​ Anal.5242014​​​‌, 1666--1691URL: http://dx.doi.org/10.1137/130913432​DOIback to text​‌
• 95 articleL.L.​​ Boltzmann. Weitere Studien​​​‌ über das Wärmegleichgewicht unter​ Gasmolekl̎en.Wiener Berichte​‌661872, 275--370​​back to text
• 96​​​‌ articleY.Yann Brenier​. Décomposition polaire et​‌ réarrangement monotone des champs​​ de vecteurs.CR​​​‌ Acad. Sci. Paris Sér.​ I Math.3051987​‌, 805--808back to​​ text
• 97 articleD.​​​‌D. Burini, S.​S. De Lillo and​‌ L.L. Gibelli.​​ Collective learning modeling based​​ on the kinetic theory​​​‌ of active particles.‌Phys. Life Rev.16‌​‌2016, 123--139back​​ to text
• 98 article​​​‌Y.Yunhai Cai.‌ Linear theory of microwave‌​‌ instability in electron storage​​ rings.Physical Review​​​‌ Special Topics-Accelerators and Beams‌1462011,‌​‌ 061002back to text​​back to text
• 99​​​‌ articleJ.J. Cai‌ and J.J. Shen‌​‌. Two classes of​​ linearly implicit local energy-preserving​​​‌ approach for general multi-symplectic‌ Hamiltonian PDEs.401‌​‌2020, 108975back​​ to text
• 100 article​​​‌C.Clément Cancès,‌ C.Claire Chainais-Hillairet,‌​‌ J.Jürgen Fuhrmann and​​ B.Benôit Gaudeul.​​​‌ A numerical analysis focused‌ comparison of several Finite‌​‌ Volume schemes for an​​ Unipolar Degenerated Drift-Diffusion Model​​​‌.IMA Journal of‌ Numerical Analysis411‌​‌2021, 271-314HAL​​DOIback to text​​​‌
• 101 articleC.Clément‌ Cancès, C.Claire‌​‌ Chainais-Hillairet, A.Anita​​ Gerstenmayer and A.Ansgar​​​‌ Jüngel. Convergence of‌ a Finite-Volume Scheme for‌​‌ a Degenerate Cross-Diffusion Model​​ for Ion Transport.​​​‌Numerical Methods for Partial‌ Differential Equations352‌​‌https://arxiv.org/abs/1801.094082019, 545-575​​HALDOIback to​​​‌ text
• 102 articleC.‌Clément Cancès, C.‌​‌Claire Chainais-Hillairet, M.​​Maxime Herda and S.​​​‌Stella Krell. Large‌ time behavior of nonlinear‌​‌ finite volume schemes for​​ convection-diffusion equations.SIAM​​​‌ Journal on Numerical Analysis‌585September 2020‌​‌, 2544-2571HALDOI​​back to text
• 103​​​‌ articleC.Clément Cancès‌, C.Claire Chainais-Hillairet‌​‌ and S.Stella Krell​​. Numerical analysis of​​​‌ a nonlinear free-energy diminishing‌ Discrete Duality Finite Volume‌​‌ scheme for convection diffusion​​ equations.Computational Methods​​​‌ in Applied Mathematics18‌3https://arxiv.org/abs/1705.10558 - Special‌​‌ issue on ''Advanced numerical​​ methods: recent developments, analysis​​​‌ and application''2018,‌ 407-432HALDOIback‌​‌ to text
• 104 article​​C.Clément Cancès,​​​‌ C.Claire Chainais-Hillairet,‌ B.Benôit Merlet,‌​‌ F.Federica Raimondi and​​ J.Juliette Venel.​​​‌ Mathematical analysis of a‌ thermodynamically consistent reduced model‌​‌ for iron corrosion.​​Zeitschrift für Angewandte Mathematik​​​‌ und Physik = Journal‌ of Applied mathematics and‌​‌ physics = Journal de​​ mathématiques et de physique​​​‌ appliquées743June‌ 2023, 96HAL‌​‌DOIback to text​​
• 105 articleC.Clément​​​‌ Cancès, T.Thomas‌ Gallouët, M.Maxime‌​‌ Laborde and L.Léonard​​ Monsaingeon. Simulation of​​​‌ multiphase porous media flows‌ with minimizing movement and‌​‌ finite volume schemes.​​European Journal of Applied​​​‌ Mathematics3062019‌, 1123-1152HALDOI‌​‌back to text
• 106​​ articleC.Clément Cancès​​​‌, T.Thomas Gallouët‌ and G.Gabriele Todeschi‌​‌. A variational finite​​ volume scheme for Wasserstein​​​‌ gradient flows.Numerische‌ Mathematik14632020‌​‌, pp 437 -​​ 480HALDOIback​​​‌ to text
• 107 article‌C.Clément Cancès and‌​‌ B.Benôit Gaudeul.​​ A convergent entropy diminishing​​​‌ finite volume scheme for‌ a cross-diffusion system.‌​‌SIAM Journal on Numerical​​ Analysis585https://arxiv.org/abs/2001.11222​​​‌2020, pp. 2784-2710‌HALDOIback to‌​‌ text
• 108 articleC.​​​‌Clément Cancès and F.​Flore Nabet. Finite​‌ Volume approximation of a​​ two-phase two fluxes degenerate​​​‌ Cahn-Hilliard model.ESAIM:​ Mathematical Modelling and Numerical​‌ Analysis5532021​​, 969--1003HALDOI​​​‌back to text
• 109​ articleC.Clément Cancès​‌, F.Flore Nabet​​ and M.Martin Vohral\'ik​​​‌. Convergence and a​ posteriori error analysis for​‌ energy-stable finite element approximations​​ of degenerate parabolic equations​​​‌.Mathematics of Computation​903282021,​‌ 517-563HALDOIback​​ to text
• 110 article​​​‌C.Clément Cancès and​ A.Antoine Zurek.​‌ A convergent finite volume​​ scheme for dissipation driven​​​‌ models with volume filling​ constraint.Numerische Mathematik​‌1512022, 279--328​​HALDOIback to​​​‌ textback to text​
• 111 incollectionJ.José​‌ Carrillo, J.Jingwei​​ Hu, Z.Zheng​​​‌ Ma and T.Thomas​ Rey. Recent development​‌ in kinetic theory of​​ granular materials: analysis and​​​‌ numerical methods.Trails​ in Kinetic TheorySEMA​‌ SIMAI Springer Serieshttps://arxiv.org/abs/2001.11206​​ - 36 pages, 5​​​‌ figuresSpringerFebruary 2021​, 1-36HALDOI​‌back to text
• 112​​ articleJ. A.J.​​​‌ A. Carrillo, A.​A. Jüngel, P.​‌ A.P. A. Markowich​​, G.G. Toscani​​​‌ and A.A. Unterreiter​. Entropy dissipation methods​‌ for degenerate parabolic problems​​ and generalized Sobolev inequalities​​​‌.Monatsh. Math.133​12001, 1--82​‌URL: http://dx.doi.org/10.1007/s006050170032DOIback​​ to text
• 113 article​​​‌C.C. Chainais-Hillairet and​ F.F. Filbet.​‌ Asymptotic behaviour of a​​ finite-volume scheme for the​​​‌ transient drift-diffusion model.​IMA J. Numer. Anal.​‌2742007,​​ 689--716URL: http://dx.doi.org/10.1093/imanum/drl045DOI​​​‌back to text
• 114​ articleC.Claire Chainais-Hillairet​‌, M.Maxime Herda​​, S.Simon Lemaire​​​‌ and J.Julien Moatti​. Long-time behaviour of​‌ hybrid finite volume schemes​​ for advection-diffusion equations: linear​​​‌ and nonlinear approaches.​Numerische Mathematik1512022​‌, 963-1016HALDOI​​back to textback​​​‌ to text
• 115 article​C.C. Chainais-Hillairet,​‌ A.A. Jüngel and​​ S.S. Schuchnigg.​​​‌ Entropy-dissipative discretization of nonlinear​ diffusion equations and discrete​‌ Beckner inequalities.Modelisation​​ Mathématique et Analyse Numérique​​​‌5012016,​ 135-162HALback to​‌ text
• 116 articleC.​​Claire Chainais-Hillairet, B.​​​‌Benôit Merlet and A.​Antoine Zurek. Convergence​‌ of a finite volume​​ scheme for a parabolic​​​‌ system with a free​ boundary modeling concrete carbonation​‌.ESAIM: Mathematical Modelling​​ and Numerical Analysis52​​​‌22018, 457-480​HALDOIback to​‌ text
• 117 articleT.​​Théophile Chaumont-Frelet, A.​​​‌Alexandre Ern, S.​Simon Lemaire and F.​‌Frédéric Valentin. Bridging​​ the Multiscale Hybrid-Mixed and​​​‌ Multiscale Hybrid High-Order methods​.ESAIM: Mathematical Modelling​‌ and Numerical Analysis56​​12022, 261-285​​​‌HALDOIback to​ text
• 118 articleF.​‌Florent Chave, D.​​ A.Daniele Antonio Di​​​‌ Pietro and S.Simon​ Lemaire. A discrete​‌ Weber inequality on three-dimensional​​ hybrid spaces with application​​​‌ to the HHO approximation​ of magnetostatics.Mathematical​‌ Models and Methods in​​ Applied Sciences321​​2022, 175-207HAL​​​‌DOIback to text‌
• 119 articleM.Matteo‌​‌ Cicuttin, A.Alexandre​​ Ern and S.Simon​​​‌ Lemaire. A Hybrid‌ High-Order method for highly‌​‌ oscillatory elliptic problems.​​Computational Methods in Applied​​​‌ Mathematics1942019‌, 723-748HALDOI‌​‌back to text
• 120​​ articleB.B. Cockburn​​​‌, J.J. Gopalakrishnan‌ and R.R. Lazarov‌​‌. Unified hybridization of​​ discontinuous Galerkin, mixed, and​​​‌ continuous Galerkin methods for‌ second-order elliptic problems.‌​‌SIAM J. Numer. Anal.​​4722009,​​​‌ 1319--1365back to text‌
• 121 bookO.O.‌​‌ Coussy. Poromechanics.​​John Wiley & Sons​​​‌2004back to text‌
• 122 inproceedingsM.Marco‌​‌ Cuturi. Sinkhorn Distances:​​ Lightspeed Computation of Optimal​​​‌ Transport.Proceedings of‌ the 26th International Conference‌​‌ on Neural Information Processing​​ Systems - Volume 2​​​‌NIPS'13Red Hook, NY,‌ USALake Tahoe, Nevada‌​‌Curran Associates Inc.2013​​, 2292–2300back to​​​‌ text
• 123 articleD.‌ ..D .A. Di‌​‌ Pietro and A.A.​​ Ern. A Hybrid​​​‌ High-Order locking-free method for‌ linear elasticity on general‌​‌ meshes.Comput. Methods​​ Appl. Mech. Engrg.283​​​‌2015, 1--21back‌ to text
• 124 article‌​‌A.A. Dragulescu and​​ V. M.V. M.​​​‌ Yakovenko. Statistical mechanics‌ of money.The‌​‌ European Physical Journal B-Condensed​​ Matter and Complex Systems​​​‌1742000,‌ 723--729back to text‌​‌
• 125 articleJ.J.​​ Droniou. Finite volume​​​‌ schemes for diffusion equations:‌ introduction to and review‌​‌ of modern methods.​​Math. Models Methods Appl.​​​‌ Sci.2482014‌, 1575-1620back to‌​‌ text
• 126 articleC.​​C Evain, C.​​​‌C Szwaj, E.‌E Roussel, J.‌​‌J Rodriguez, M.​​M Le Parquier,​​​‌ M.-A.M-A Tordeux,‌ F.F Ribeiro,‌​‌ M.M Labat,​​ N.N Hubert,​​​‌ J.-B.J-B Brubach and‌ others. Stable coherent‌​‌ terahertz synchrotron radiation from​​ controlled relativistic electron bunches​​​‌.Nature Physics15‌72019, 635--639‌​‌back to text
• 127​​ articleR.R. Eymard​​​‌, T.T. Gallouët‌, C.C. Guichard‌​‌, R.R. Herbin​​ and R.R. Masson​​​‌. TP or not‌ TP, that is the‌​‌ question.Comput. Geosci.​​182014, 285--296​​​‌back to text
• 128‌ articleT.Thomas Gallouët‌​‌, Q.Quentin Merigot​​ and A.Andrea Natale​​​‌. Convergence of a‌ Lagrangian discretization for barotropic‌​‌ fluids and porous media​​ flow.SIAM Journal​​​‌ on Mathematical Analysis54‌32022HALDOI‌​‌back to text
• 129​​ inproceedingsK.K. Gärtner​​​‌ and L.L. Kamenski‌. Why Do We‌​‌ Need Voronoi Cells and​​ Delaunay Meshes?Numerical Geometry,​​​‌ Grid Generation and Scientific‌ ComputingLecture Notes in‌​‌ Computational Science and Engineering​​ChamSpringer International Publishing​​​‌2019, 45--60DOI‌back to text
• 130‌​‌ articleB.Benôit Gaudeul​​ and J.Jürgen Fuhrmann​​​‌. Entropy and convergence‌ analysis for two finite‌​‌ volume schemes for a​​ Nernst-Planck-Poisson system with ion​​​‌ volume constraints.Numerische‌ Mathematik1511April‌​‌ 2022, 99-149HAL​​​‌DOIback to text​
• 131 articleS.S.​‌ Gottlieb, C.-W.C.-W.​​ Shu and E.E.​​​‌ Tadmor. Strong stability-preserving​ high-order time discretization methods​‌.SIAM Rev.43​​12001, 89--112​​​‌URL: https://doi.org/10.1137/S003614450036757Xback to​ text
• 132 articleM.​‌Maxime Herda and L.​​ M.Luis Miguel Miguel​​​‌ Rodrigues. Anisotropic Boltzmann-Gibbs​ dynamics of strongly magnetized​‌ Vlasov-Fokker-Planck equations.Kinetic​​ and Related Models 12​​​‌3https://arxiv.org/abs/1610.051382019,​ 593-636HALDOIback​‌ to text
• 133 article​​P.-E.Pierre-Emmanuel Jabin and​​​‌ T.Thomas Rey.​ Hydrodynamic limit of granular​‌ gases to pressureless Euler​​ in dimension 1.​​​‌Quarterly of Applied Mathematics​75https://arxiv.org/abs/1602.09103 - 26​‌ pages, 1 figure2017​​, 155-179HALDOI​​​‌back to text
• 134​ articleR.R. Jordan​‌, D.D. Kinderlehrer​​ and F.F. Otto​​​‌. The variational formulation​ of the Fokker-Planck equation​‌.SIAM J. Math.​​ Anal.2911998​​​‌, 1--17back to​ text
• 135 articleL.​‌L. Kantorovitch. On​​ the translocation of masses​​​‌.C. R. (Dokl.)​ Acad. Sci. URSS, n.​‌ Ser.371942,​​ 199--201back to text​​​‌
• 136 articleC.C.​ L eBris, F.​‌F. Legoll and A.​​A. Lozinski. MsFEM​​​‌ à la Crouzeix--Raviart for​ highly oscillatory elliptic problems​‌.Chinese Annals of​​ Mathematics, Series B34​​​‌12013, 113--138​back to text
• 137​‌ articleP.P. Lafitte​​ and G.G. Samaey​​​‌. Asymptotic-preserving Projective Integration​ Schemes for Kinetic Equations​‌ in the Diffusion Limit​​.3422012​​​‌, A579–-A602back to​ text
• 138 articleA.​‌ M.A. M. M.​​ Leal, D. A.​​​‌D. A. Kulik,​ W. R.W. R.​‌ Smith and M. O.​​M. O. Saar.​​​‌ An overview of computational​ methods for chemical equilibrium​‌ and kinetic calculations for​​ geochemical and reactive transport​​​‌ modeling.Pure Appl.​ Chem.8952017​‌, 597--643back to​​ text
• 139 articleS.​​​‌Simon Lemaire. Bridging​ the hybrid high-order and​‌ virtual element methods.​​IMA J. Numer. Anal.​​​‌4112021,​ 549--593DOIback to​‌ text
• 140 articleD.​​D. Matthes and S.​​​‌S. Plazzotta. A​ variational formulation of the​‌ BDF2 method for metric​​ gradient flows.ESAIM:​​​‌ M2AN5312019​, 145-172back to​‌ text
• 141 articleA.​​A. Mielke. A​​​‌ gradient structure for reaction-diffusion​ systems and for energy-drift-diffusion​‌ systems.Nonlinearity24​​42011, 1329--1346​​​‌URL: http://dx.doi.org/10.1088/0951-7715/24/4/016DOIback​ to text
• 142 article​‌J.JB Murphy,​​ R.RL Gluckstern and​​​‌ S.S Krinsky.​ Longitudinal wake field for​‌ an electron moving on​​ a circular orbit.​​​‌Part. Accel.57BNL-63090​1996, 9--64back​‌ to text
• 143 article​​A.Andrea Natale and​​​‌ G.Gabriele Todeschi.​ A mixed finite element​‌ discretization of dynamical optimal​​ transport.Journal of​​​‌ Scientific Computing912​2022, 1--26back​‌ to text
• 144 article​​A.Andrea Natale and​​​‌ G.Gabriele Todeschi.​ Computation of optimal transport​‌ with finite volumes.​​ESAIM: Mathematical Modelling and​​ Numerical Analysis555​​​‌September 2021, 1847-1871‌HALDOIback to‌​‌ textback to text​​
• 145 inproceedingsA.Andrea​​​‌ Natale and G.Gabriele‌ Todeschi. TPFA finite‌​‌ volume approximation of Wasserstein​​ gradient flows.International​​​‌ Conference on Finite Volumes‌ for Complex ApplicationsSpringer‌​‌2020, 193--201back​​ to text
• 146 article​​​‌L.L. Onsager.‌ Reciprocal relations in irreversible‌​‌ processes. II..Physical​​ Review381931,​​​‌ 2265--2279back to text‌
• 147 articleM. A.‌​‌Mark A. Peletier and​​ M.Matthias Röger.​​​‌ Partial Localization, Lipid Bilayers,‌ and the Elastica Functional‌​‌.1933September​​ 2009, 475--537DOI​​​‌back to text
• 148‌ unpublishedM. A.M.‌​‌ A. Peletier. Variational​​ Modelling: Energies, gradient flows,​​​‌ and large deviations.‌2014, arXiv:1402.1990back‌​‌ to text
• 149 article​​G.Gabriel Peyré,​​​‌ M.Marco Cuturi and‌ others. Computational optimal‌​‌ transport.Center for​​ Research in Economics and​​​‌ Statistics Working Papers2017-86‌2017back to text‌​‌
• 150 phdthesisE.Eléonore​​ Roussel. Spatio-temporal dynamics​​​‌ of relativistic electron bunches‌ during the microbunching instability‌​‌ : study of the​​ Synchrotron SOLEIL and UVSOR​​​‌ storage rings.Université‌ Lille1 - Sciences et‌​‌ TechnologiesSeptember 2014HAL​​back to text
• 151​​​‌ bookF.Filippo Santambrogio‌. Optimal transport for‌​‌ applied mathematicians.Birkäuser,​​ NY5558-63Springer​​​‌2015, 94back‌ to text
• 152 article‌​‌J.J. Saragosti,​​ V.V. Calvez,​​​‌ N.N. Bournaveas,‌ B.B. Perthame,‌​‌ A.A. Buguin and​​ P.P. Silberzan.​​​‌ Directional persistence of chemotactic‌ bacteria in a traveling‌​‌ concentration wave.Proceedings​​ of the National Academy​​​‌ of Sciences10839‌2011, 16235--16240back‌​‌ to text
• 153 article​​B.B. Seguin and​​​‌ N. J.N. J.‌ Walkington. Multi-Component Multiphase‌​‌ Flow Through a Poroelastic​​ Medium.J. Elasticity​​​‌1352019, 485-507‌back to text
• 154‌​‌ articleB.B. Seguin​​ and N. J.N.​​​‌ J. Walkington. Multi-component‌ Multiphase Porous Flow.‌​‌Arch. Ration. Mech. Anal.​​2352020, 2171-2196​​​‌URL: https://doi.org/10.1007/s00205-019-01473-7back to‌ text
• 155 articleF.‌​‌Filippo Simini, M.​​ C.Marta C González​​​‌, A.Amos Maritan‌ and A.-L.Albert-László Barabási‌​‌. A universal model​​ for mobility and migration​​​‌ patterns.Nature484‌73922012, 96--100‌​‌back to text
• 156​​ unpublishedF.F. Sma\"i​​​‌. A thermodynamic formulation‌ for multiphase compositional flows‌​‌ in porous media.​​2020, HAL: hal-02925433​​​‌back to text