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 118, the multiphase and multicomponent character of the fluid which flows therein 82, and chemical reactions with a wide range of characteristic times 137. 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.

The kinetic theory of molecular gases models a gas as a system of elastically colliding spheres, conserving mechanical energy during impact. Once initialized, it takes to a molecular gas no more than a few collisions per particle to relax to its equilibrium state, characterized by a Maxwellian velocity distribution and a certain homogeneous density (in the absence of external forces). A granular gas is a system of dissipatively colliding, macroscopic particles (grains). This slight change in the microscopic dynamics (converting energy into heat) causes drastic changes in the behavior of the gas: granular gases are open systems, which exhibit self-organized spatio-temporal cluster formations, and have no equilibrium distribution. They can be used to model silos, avalanches, pollen or planetary rings.

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 75. We are interested in the devising of HHO methods in the curl/curl setting 115. 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, 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 94, 123.

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 94 with a well suited mean-field self-interaction term 144. 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.

5.1 New software

6.1 Modeling, analysis and numerical simulation of multi-component systems

Chemical equilibria computations, especially those with vanishing species in the aqueous phase, lead to nonlinear systems that are difficult to solve due to gradient blow up. Instead of the commonly used ad hoc treatments, M. Jonval, C. Cancès et al. propose in 33 two reformulations of the single-phase chemical equilibrium problem which are in line with the spirit of preconditioning but whose actual aims are to guarantee a better stability of Newton's method. The first reformulation is a parametrization of the graph linking species mole fractions to their chemical potentials. The second is based on an augmented system where this relationship is relaxed for the iterates by means of a Cartesian representation. We theoretically prove the local quadratic convergence of Newton's method for both reformulations. From a numerical point of view, we demonstrate that the two techniques are accurate, allowing to compute equilibria with chemical species having very low concentrations. Moreover, the robustness of our methods combined with a globalization strategy is superior to that of the literature.

In 13, C. Chainais-Hillairet, M. Herda et al. consider a drift-diffusion charge transport model for perovskite solar cells, where electrons and holes may diffuse linearly (Boltzmann approximation) or nonlinearly (e.g. due to Fermi-Dirac statistics). To incorporate volume exclusion effects, they rely on the Fermi-Dirac integral of order $-1$ when modeling moving anionic vacancies within the perovskite layer which is sandwiched between electron and hole transport layers. After non-dimensionalization, they first prove a continuous entropy-dissipation inequality for the model. Then, they formulate a corresponding two-point flux finite volume scheme on Voronoi meshes and show an analogous discrete entropy-dissipation inequality. This inequality helps them to show the existence of a discrete solution to the nonlinear discrete system with the help of a corollary of Brouwer's fixed point theorem and the minimization of a convex functional. Finally, they verify the theoretically proven properties numerically, simulating a realistic device setup and showing exponential decay in time with respect to the $L^2$-error as well as a physically and analytically meaningful relative entropy.

In 38, C. Chainais-Hillairet, M. Herda 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 as components for next-generation artificial intelligence (AI) hardware and information technology. The 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 non-negativity of carrier densities. Novel discrete entropy-dissipation inequalities for both boundary condition types in multiple dimensions allow them 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 62, M. Herda et al. analyse an instationary drift–diffusion system for the electron, hole, and oxygen vacancy densities, coupled to the Poisson equation for the electric potential, in a bounded domain with mixed Dirichlet–Neumann boundary conditions. The electron and hole densities are governed by Fermi–Dirac statistics, while the oxygen vacancy density is governed by Blakemore statistics. The equations model the charge carrier dynamics in memristive devices used in semiconductor technology. The global existence of weak solutions is proved in up to three space dimensions. The proof is based on the free energy inequality, an iteration argument to improve the integrability of the densities, and estimations of the Fermi–Dirac integral. Under a physically realistic elliptic regularity condition, it is proved that the densities are bounded.

In 17, C. Cancès, C. Chainais-Hillairet et al. propose and study a one-dimensional model which consists of two cross-diffusion systems coupled via a moving interface. The motivation stems from the modelling of complex diffusion processes in the context of the vapor deposition of thin films. In the model, cross-diffusion of the various chemical species can be respectively modelled by a size-exclusion system for the solid phase and the Stefan-Maxwell system for the gaseous phase. The coupling between the two phases is modelled by linear phase transition laws of Butler-Volmer type, resulting in an interface evolution. The continuous properties of the model are investigated, in particular its entropy variational structure and stationary states. They introduce a two-point flux approximation finite volume scheme. The moving interface is addressed with a moving-mesh approach, where the mesh is locally deformed around the interface. The resulting discrete nonlinear system is shown to admit a solution that preserves the main properties of the continuous system, namely: mass conservation, nonnegativity, volume-filling constraints, decay of the free energy and asymptotics. In particular, the moving-mesh approach is compatible with the entropy structure of the continuous model. Numerical results illustrate these properties and the dynamics of the model.

In 42, C. Cancès, M. Herda et al. present a finite volume scheme for modeling the diffusion of charged particles, specifically ions, in constrained geometries using a degenerate Poisson-Nernst-Planck system with size exclusion yielding cross-diffusion. The method utilizes a two-point flux approximation and is part of the exponentially fitted schemes framework. The scheme is shown to be thermodynamically consistent, as it ensures the decay of some discrete version of the free energy. Classical numerical analysis results –existence of discrete solution, convergence of the scheme as the grid size and the time step go to 0– follow. They also investigate the long-time behavior of the scheme, both from a theoretical and numerical point of view. Numerical simulations confirm the findings, but also point out some possibly very slow convergence towards equilibrium of the system under consideration.

In 43, C. Cancès et al. study a system of drift-diffusion PDEs for a potentially infinite number of incompressible phases, subject to a joint pointwise volume constraint. The analysis is based on the interpretation as a collection of coupled Wasserstein gradient flows or, equivalently, as a gradient flow in the space of couplings under a 'fibered' Wasserstein distance. They prove existence of weak solutions, long-time asymptotics, and stability with respect to the mass distribution of the phases, including the discrete to continuous limit. A key step is to establish convergence of the product of pressure gradient and density, jointly over the infinite number of phases. The underlying energy functional is the objective of entropy regularized optimal transport, which allows us to interpret the model as the relaxation of the classical Angenent-Haker-Tannenbaum (AHT) scheme to the entropic setting. However, in contrast to the AHT scheme's lack of convergence guarantees, the relaxed scheme is unconditionally convergent. They conclude with numerical illustrations of the main results.

Multiphase poromechanics describes the evolution of multiphase flows in deformable porous media. Mathematical models for such multiphysics systems are inherently nonlinear, potentially degenerate and fully coupled systems of partial differential equations. In 40, C. Cancès et al. present a thermodynamically consistent multiphase poromechanics model falling into the category of Biot equations and obeying to a generalized gradient flow structure. It involves capillarity effects, degenerate relative permeabilities, and gravity effects. Contrary to established models it introduces a Lagrange multiplier associated to a bound constraint on the effective porosity in particular ensuring its positivity. They establish existence of global weak solutions under the assumption of a weak coupling strength, implicitly utilizing the gradient flow structure, as well as regularization, a Faedo-Galerkin approach, and compactness arguments. This comprises the first global existence result for multiphase poromechanics accounting for degeneracies that are consistent with the multiphase nature of the flow.

The presence of moving interfaces is a characteristic feature of corrosion phenomena. The evolution of these interfaces is governed by nonlinear laws and poses significant difficulties in the theoretical and numerical treatment of the models. B. Merlet, J. Venel et al. attack these difficulties in 64, namely, they show the existence of solutions to a 1D reduced corrosion model with a moving interface. Their method is based on a minimizing movement scheme for some modified Wasserstein metric.

The efficiency of (In,Ga)N-based light emitting diodes (LEDs) is limited by the failure of holes to evenly distribute across the (In,Ga)N/GaN multi-quantum well stack which forms the active region. To tackle this problem, it is important to understand carrier transport in these alloys. In 37, J. Moatti et al. study the impact that random alloy fluctuations have on the distribution of electrons and holes in such devices. To do so, an atomistic tight-binding model is employed to account for alloy fluctuations on a microscopic level and the resulting tight-binding energy landscape forms input to a quantum corrected drift-diffusion model. Here, quantum corrections are introduced via localization landscape theory. Similar to experimental studies in the literature, they have focused on a multi-quantum well system where two of the three wells have the same In content while the third well differs in In content. By changing the order of wells in this ‘multi-color’ quantum well structure and looking at the relative radiative recombination rates of the different emitted wavelengths, they (i) gain insight into the distribution of carriers in such a system and (ii) can compare their findings to trends observed in experiment. They focus on three factors and evaluate the impact that each have on carrier distribution: an electron blocking layer, quantum corrections and random alloy fluctuations. They find that the electron blocking layer is of secondary importance. However, in order to recover experimentally observed features – namely that the p-side quantum well dominates the light emission – both quantum corrections and random alloy fluctuations should be considered. The widely assumed homogeneous virtual crystal approximation fails to capture the characteristic light emission distribution across a multi-quantum well stack.

6.2 Analysis and numerical simulation in electromagnetism and related fields

In 36, S. Lemaire and S. Pitassi prove discrete versions of the first and second Weber inequalities on $H\left(\mathrm{𝐜𝐮𝐫𝐥}\right)\cap H\left(\mathrm{div}\right)$-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 $H\left(\mathrm{𝐜𝐮𝐫𝐥}\right)$- and $H\left(\mathrm{div}\right)$-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 are also proved, within a general topological setting, related discrete Maxwell compactness properties.

In 56, S. Lemaire, S. Pitassi et al. 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 systems they focus on stem from magnetostatics, and can either be first-order (field formulation) or second-order (vector potential formulation). The well-posedness of the methods essentially relies on topologically generic hybrid versions of the first and second Weber inequalities. The error analysis is performed for regular solutions. Finally, leveraging (co)homology computation techniques from the literature, a comprehensive numerical validation of the methodology is provided.

In 50, T. Chaumont-Frelet et al. analyze the conforming approximation of the time-harmonic Maxwell's equations using Nédélec (edge) finite elements. They prove that the approximation is asymptotically optimal, i.e., the approximation error in the energy norm is bounded by the best-approximation error times a constant that tends to one as the mesh is refined and/or the polynomial degree is increased. Moreover, under the same conditions on the mesh and/or the polynomial degree, they establish discrete inf-sup stability with a constant that corresponds to the continuous constant up to a factor of two at most. Their proofs apply under minimal regularity assumptions on the exact solution, so that general domains, material coefficients, and right-hand sides are allowed.

In 47, T. 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 48, T. Chaumont-Frelet considers interior penalty discontinuous Galerkin discretizations of time-harmonic wave propagation problems modeled by the Helmholtz equation, and derives novel a priori and a posteriori estimates. His analysis classically relies on duality arguments of Aubin–Nitsche type, and its originality is that it applies under minimal regularity assumptions. The estimates he obtains directly generalize known results for conforming discretizations, namely that the discrete solution is optimal in a suitable energy norm and that the error can be explicitly controlled by a posteriori estimators, provided the mesh is sufficiently fine.

In 49, T. Chaumont-Frelet et al. derive a priori and a posteriori error estimates for the discontinuous Galerkin (dG) approximation of the time-harmonic Maxwell's equations. Specifically, they consider an interior penalty dG method, and establish error estimates that are valid under minimal regularity assumptions and involving constants that do not depend on the frequency for sufficiently fine meshes. The key result of the a priori error analysis is that the dG solution is asymptotically optimal in an augmented energy norm that contains the dG stabilization. Specifically, up to a constant that tends to one as the mesh is refined, the dG solution is as accurate as the best approximation in the same norm. The main insight is that the quantities controlling the smallness of the mesh size are essentially those already appearing in the conforming setting. They also show that for fine meshes, the inf-sup stability constant is as good as the continuous one up to a factor two. Concerning the a posteriori analysis, they consider a residual-based error estimator under the assumption of piecewise constant material properties. They derive a global upper bound and local lower bounds on the error with constants that (i) only depend on the shape-regularity of the mesh if it is sufficiently refined and (ii) are independent of the stabilization bilinear form.

In 55, T. Chaumont-Frelet et al. consider the $h$-version of the finite element method, where accuracy is increased by decreasing the meshwidth $h$ while keeping the polynomial degree $p$ constant, applied to the Helmholtz equation. Although the question "how quickly must $h$ decrease as the wavenumber $k$ increases to maintain accuracy?" has been studied intensively since the 1990s, none of the existing rigorous wavenumber-explicit analyses take into account the approximation of the geometry. In this paper they prove that for nontrapping problems solved using straight elements the geometric error is of order $kh$, which is then less than the pollution error $k{\left(kh\right)}^{2}p$ when $k$ is large; this fact is then illustrated in numerical experiments. More generally, they prove that, even for problems with strong trapping, using degree four (in 2-d) or degree five (in 3-d) polynomials and isoparametric elements ensures that the geometric error is smaller than the pollution error for most large wavenumbers.

In 51, T. Chaumont-Frelet et al. derive a posteriori error estimates for the the scalar wave equation discretized in space by continuous finite elements and in time by the explicit leapfrog scheme. The analysis combines the idea of invoking extra time-regularity for the right-hand side, as previously introduced in the space semi-discrete setting, with a novel, piecewise quartic, globally twice-differentiable time-reconstruction of the fully discrete solution. The main results show that the proposed estimator is reliable and efficient in a damped energy norm. These properties are illustrated in a series of numerical examples.

In 52, T. Chaumont-Frelet et al. prove sharp wavenumber-explicit error bounds for first- or second-type-Nédélec-element (a.k.a. edge-element) conforming discretisations, of arbitrary (fixed) order, of the variable-coefficient time-harmonic Maxwell equations posed in a bounded domain with perfect electric conductor (PEC) boundary conditions. The PDE coefficients are allowed to be piecewise regular and complex-valued; this set-up therefore includes scattering from a PEC obstacle and/or variable real-valued coefficients, with the radiation condition approximated by a perfectly matched layer (PML). In the analysis of the $h$-version of the finite element method, with fixed polynomial degree $p$, applied to the time-harmonic Maxwell equations, the asymptotic regime is when the meshwidth, $h$, is small enough (in a wavenumber-dependent way) that the Galerkin solution is quasioptimal independently of the wavenumber, while the preasymptotic regime is the complement of the asymptotic regime. The results of this paper are the first preasymptotic error bounds for the time-harmonic Maxwell equations using first-type Nédélec elements or higher-than-lowest-order second-type Nédélec elements. Furthermore, they are the first wavenumber-explicit results, even in the asymptotic regime, for Maxwell scattering problems with a non-empty scatterer.

In 66, T. Chaumont-Frelet et al. consider the finite element solution of time-harmonic wave propagation problems in heterogeneous media with hybridizable discontinuous Galerkin (HDG) methods. In the case of homogeneous media, it has been observed that the iterative solution of the linear system can be accelerated by hybridizing with transmission variables instead of numerical traces, as performed in standard approaches. In this work, they extend the HDG method with transmission variables, which is called the CHDG method, to the heterogeneous case with piecewise constant physical coefficients. In particular, they consider formulations with standard upwind and general symmetric fluxes. The CHDG hybridized system can be written as a fixed-point problem, which can be solved with stationary iterative schemes for a class of symmetric fluxes. The standard HDG and CHDG methods are systematically studied with the different numerical fluxes by considering a series of 2D numerical benchmarks. The convergence of standard iterative schemes is always faster with the extended CHDG method than with the standard HDG methods, with upwind and scalar symmetric fluxes.

In 25, E. Creusé et al. propose an a posteriori goal-oriented error estimator for the harmonic ($A$–$\Phi$) formulation arising in the modeling of eddy current problems, approximated by nonconforming finite element methods. It is based on the resolution of a dual problem associated with the initial one. For each of these two problems, a guaranteed equilibrated estimator is developed using some flux reconstructions. These fluxes also allow to obtain a goal-oriented error estimator that is fully computable and can be split in a principal part and a remainder one. Their theoretical results are illustrated by numerical experiments.

In 54, T. Chaumont-Frelet et al. consider time-harmonic elastodynamic problems in heterogeneous media. They focus on scattering problems in the high-frequency regime and in nearly incompressible media, where the angular frequency $\omega$ and ratio of the Lamé parameters $\lambda /\mu$ may both be large. They derive stability estimates controlling the norm of the solution by the norm of the right-hand side up to a fully-explicit constant. Crucially, under natural assumptions on the domain and coefficients, this constant increases linearly with $\omega$ and is uniform in the ratio $\lambda /\mu$.

6.3 Analysis of dissipative models and their discretization

In 63, M. 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 gradient flow structure 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, non-negativity of solutions, entropy, and duality estimates. These properties allow them to prove the well-posedness, 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 45, C. Cancès, A. Natale et al. propose a fully discrete finite volume scheme for the standard Fokker–Planck equation. The space discretization relies on the well-known square-root approximation, which falls into the framework of two-point flux approximations. The time discretization is novel and relies on a tailored nonlinear mid-point rule, designed to accurately capture the dissipative structure of the model. They establish well-posedness for the scheme, positivity of the solutions, as well as a fully discrete energy-dissipation inequality mimicking the continuous one. They then prove the convergence of the scheme under mildly restrictive conditions on the unstructured grids, which can be easily satisfied in practice. Numerical simulations show that the scheme is second order accurate both in time and space, and that one can solve the discrete nonlinear systems arising at each time step using Newton's method with low computational cost.

In 39, P. Gervais et al. prove a criterion justifying propagation and appearance of $L^p$–norms for solutions to the spatially homogeneous Boltzmann equation with very soft potentials without cutoff. Such a criterion also provides a new conditional stability and uniqueness result for classical solutions to the equation. They first derive an energy estimate, which they close thanks to a new “$ϵ$-Poincaré inequality”, allowing then for an estimate on the appearance rate of $L^p$–norms. The strategy is then refined to handle weights and stability estimates.

In 58, P. Gervais and M. Herda prove sufficient conditions for uniqueness and stability of stationary solutions to Vlasov–Fokker–Planck equation with self consistent interactions and a confining force. They first prove existence under general assumptions, and then uniqueness for interaction kernels with small negative Fourier modes. They then prove stability under the extra assumption of small imaginary Fourier modes through hypocoercive and hypoelliptic methods.

In 59, P. Gervais proves hypoelliptic estimates and averaging lemmas for kinetic equations in presence of a confining potential. This work adapts results from the work of F. Bouchut  90 and P.-E. Jabin  131 to the standard weighted framework of stability theory in confined kinetic equations.

In 15, T. Laidin et al. consider an implicit finite volume discretization of a non-linear kinetic reaction model describing the interactions of two species. They show that the discrete solutions converge exponentially fast towards the steady state. The proof relies on a small perturbation argument and the proof of a maximum principle on the non-linear system.

In 34, T. Laidin et al. construct spectral methods that allow to exactly preserve the moments of the unknown at the discrete level. The method is general and relies on a constrained minimization problem that can be solved explicitly. The robustness of the method is illustrated through various numerical experiments ranging from bounded and unbounded domains to applications to Fokker–Planck type equations.

In 41, C. Calgaro, R. Colombier and E. Creusé propose a Finite-Volume scheme on cartesian grids for the numerical approximation of some solutions to the Quantum Navier–Stokes (QNS) equations. The QNS system corresponds to a generalization of the usual Navier–Stokes system when quantum effects need to be taken into account, as in certain physical phenomena such as superfluids evolution or electron flow in semiconductors. The main difficulty lies in the presence of the Bohm potential in the momentum equation, which corresponds to a third-order term, increasing the complexity of the numerical scheme used to obtain approximate solutions. The idea is then to introduce a new variable $\nabla f\left(\rho \right)/\rho$ where the function $f$ must be specified and to add to the original third-order system an additional equation, allowing it to be reduced to a second-order one. The originality of this contribution is to propose a scheme allowing to provide some discrete BD-entropy inequalities (Bresch–Desjardins), similarly to the continuous case (see 92). The numerical scheme is based on a time splitting, where the hyperbolic step is discretized with a classical Finite-Volume scheme, whereas the parabolic step remains linear. Discrete BD-entropy inequalities are established, and numerical tests in 1D and 2D illustrate the efficiency of the scheme in several configurations.

In 21, C. Chainais-Hillairet et al. propose a new numerical two-point flux for a quasilinear convection-diffusion equation. This numerical flux is shown to be an approximation of the numerical flux derived from the solution of a two-point Dirichlet boundary value problem for the projection of the continuous flux onto the line connecting neighboring collocation points. The latter approach generalizes an idea first proposed by Scharfetter and Gummel for linear drift-diffusion equations. Convergence of the scheme is established, as well as relative entropy decay properties.

In 14, S. Lemaire et al. study the numerical approximation of sign-shifting problems of elliptic type. They fully analyze and assess the method briefly introduced in 67. The method, which is based on domain decomposition and optimization, is proved to be convergent as soon as, for a given loading, the continuous problem admits a unique solution of finite energy. Departing from the $𝚃$-coercivity approach, which relies on the use of geometrically fitted mesh families, the method works for arbitrary (interface-compliant) meshes. Moreover, it is shown convergent for a class of problems for which $𝚃$-coercivity is not applicable. A comprehensive set of test-cases complements the analysis.

In 35, S. Lemaire and J. Moatti are interested in the high-order approximation of anisotropic, potential-driven advection-diffusion models on general polytopal partitions. They 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. In contrast, numerical evidence is displayed indicating that the linear scheme violates positivity, whatever the order. Finally, they 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.

In 20, M. Herda et al. study a self-consistent Vlasov–Fokker–Planck equation which describes the longitudinal dynamics of an electron bunch in the storage ring of a synchrotron particle accelerator. They show existence and uniqueness of global classical solutions under physical hypotheses on the initial data. The proof relies on a mild formulation of the equation and hypoelliptic regularization estimates. They also address the problem of the long-time behavior of solutions. They prove the existence of steady states, called Haissinski solutions, given implicitly by a nonlinear integral equation. When the beam current (i.e. the nonlinearity) is small enough, they show uniqueness of steady state and local asymptotic nonlinear stability of solutions in appropriate weighted Lebesgue spaces. The proof is based on hypocoercivity estimates. Finally, they discuss the physical derivation of the equation and its particular asymmetric interaction potential.

Neuron models have attracted a lot of attention recently, both in mathematics and neurosciences. In 32, M. Herda et al. are interested in studying long-time and large-population emerging properties in a simplified toy model. From a mathematical perspective, this amounts to study the long-time behaviour of a degenerate reflected diffusion process. Using coupling arguments, the flow is proven to be a contraction of the Wasserstein distance for long times, which implies the exponential relaxation toward a (non-explicit) unique globally attractive equilibrium distribution. The result is extended to a McKean–Vlasov type non-linear variation of the model, when the mean-field interaction is sufficiently small. The ergodicity of the process results from a combination of deterministic contraction properties and local diffusion, the noise being sufficient to drive the system away from non-contractive domains.

In 16, C. Calgaro, C. Cancès and E. Creusé perform the convergence analysis of a finite volume scheme for a convection-diffusion equation involving a Joule effect term which was introduced in the former contribution 96 emanating from the team. The proof proposed in 16 relies on compactness arguments. In order to treat the Joule effect term, a second order discrete Gagliardo–Nirenberg inequality is established by the authors. By going beyond the (nowadays usual) framework of discrete functional inequalities involving first order discrete differential operators, this contribution seems to be a genuine novelty in the field of discrete functional analysis.

6.4 Complementary topics in numerical analysis

In 53, T. Chaumont-Frelet et al. rewrite the standard nodal virtual element method as a generalised gradient method. This re-formulation allows for computing a reliable and efficient error estimator by locally reconstructing broken fluxes and potentials. They prove the usual upper and lower bounds with constants independent of the stabilisation of the method and, under technical assumptions on the mesh, the degree of accuracy.

In 23, T. Chaumont-Frelet et al. analyze constrained and unconstrained minimization problems on patches of tetrahedra sharing a common vertex with discontinuous piecewise polynomial data of degree $p$. They show that the discrete minimizers in the spaces of piecewise polynomials of degree $p$ conforming in the $H^1$, $H\left(\mathrm{𝐜𝐮𝐫𝐥}\right)$, or $H\left(\mathrm{div}\right)$ spaces are as good as the minimizers in these entire (infinite-dimensional) Sobolev spaces, up to a constant that is independent of $p$. These results are useful in the analysis and design of finite element methods, namely for devising stable local commuting projectors and establishing local-best/global-best equivalences in a priori analysis and in the context of a posteriori error estimation. Unconstrained minimization in $H^1$ and constrained minimization in $H\left(\mathrm{div}\right)$ have been previously treated in the literature. Along with improvement of the results in the $H^1$ and $H\left(\mathrm{div}\right)$ cases, our key contribution is the treatment of the $H\left(\mathrm{𝐜𝐮𝐫𝐥}\right)$ framework. This enables us to cover the whole de Rham diagram in three space dimensions in a single setting.

In 22, T. Chaumont-Frelet et al. design an operator from the infinite-dimensional Sobolev space $H\left(\mathrm{𝐜𝐮𝐫𝐥}\right)$ to its finite-dimensional subspace formed by the Nédélec piecewise polynomials on a tetrahedral mesh that has the following properties: 1) it is defined over the entire $H\left(\mathrm{𝐜𝐮𝐫𝐥}\right)$, including boundary conditions imposed on a part of the boundary; 2) it is defined locally in a neighborhood of each mesh element; 3) it is based on simple piecewise polynomial projections; 4) it is stable in the $L^2$-norm, up to data oscillation; 5) it has optimal (local-best) approximation properties; 6) it satisfies the commuting property with its sibling operator on $H\left(\mathrm{𝐜𝐮𝐫𝐥}\right)$; 7) it is a projector, i.e., it leaves intact objects that are already in the Nédélec piecewise polynomial space. This operator can be used in various parts of numerical analysis related to the $H\left(\mathrm{𝐜𝐮𝐫𝐥}\right)$ space. The authors employ it here to establish the two following results: i) equivalence of global-best, tangential-trace-and curl-constrained, and local-best, unconstrained approximations in $H\left(\mathrm{𝐜𝐮𝐫𝐥}\right)$ including data oscillation terms; and ii) fully $h$- and $p$- (meshsize- and polynomial-degree-) optimal approximation bounds valid under the minimal Sobolev regularity only requested elementwise. As a result of independent interest, they also prove a $p$-robust equivalence of curl-constrained and unconstrained best-approximations on a single tetrahedron in the $H\left(\mathrm{𝐜𝐮𝐫𝐥}\right)$ setting, including $hp$ data oscillation terms.

6.5 Analysis and numerical simulation of variational models

Derived from the concentration-compactness principle, the concept of generalized minimizer can be used to define generalized solutions of variational problems which may have components “infinitely” distant from each other. In 46, and under mild assumptions, J. Candau-Tilh establishes existence and density estimates of generalized minimizers of perturbed isoperimetric problems. His hypotheses encapsulate a wide class of functionals including the classical, anisotropic and fractional perimeter. The perturbation term may for instance take the form of a potential, a translation invariant kernel or a nonlocal term involving the Wasserstein distance.

In 57, A. Natale et al. study a variational problem providing a way to extend for all times minimizing geodesics connecting two given probability measures, in the Wasserstein space. This is simply obtained by allowing for negative coefficients in the classical variational characterization of Wasserstein barycenters. They show that this problem admits two equivalent convex formulations: the first can be seen as a particular instance of Toland duality and the second is a barycentric optimal transport problem. They also propose an efficient numerical scheme to solve this latter formulation based on entropic regularization and a variant of Sinkhorn algorithm.

In 29, B. Merlet, M. Pegon et al. establish a $C^{1,\alpha }$-regularity theorem for almost-minimizers of the functional ${ℱ}_{\epsilon ,\gamma }=P-\gamma P_{\epsilon }$, where $\gamma \in (0,1)$, $P$ is the classical perimeter, and $P_{\epsilon }$ is a nonlocal energy converging to the perimeter as $\epsilon$ vanishes. The theorem provides a criterion for $C^{1,\alpha }$-regularity at a point of the boundary which is uniform as the parameter $\epsilon$ goes to 0. As a consequence they obtain that volume-constrained minimizers of ${ℱ}_{\epsilon ,\gamma }$ are balls for any $\epsilon$ small enough. For small $\epsilon$, this minimization problem corresponds to the large mass regime for a Gamow-type problem where the nonlocal repulsive term is given by an integrable kernel $G$ with sufficiently fast decay at infinity.

In 44, C. Cancès et al. consider the convergence of a finite element discretization of a degenerate parabolic equation of $q$-Laplace type with an additional external potential. The main novelty of the approach presented therein is that the authors use the underlying gradient flow structure in the $L^p$-Wasserstein metric: from the abstract machinery of metric gradient flows, the convergence of the scheme is obtained solely on the basis of estimates that result naturally from the equation's variational structure. In particular, the limit is identified as the unique gradient flow solution without reference to monotonicity methods.

In 31, B. Merlet (former member of the team) et al. introduce and study the notion of tensor rectifiable chains which form a two-dimensional chain complex of normed groups. They complete the theory of tensor flat chains in 60 by establishing a deformation theorem in this setting and by identifying isometrically some subgroups of tensor flat chains with corresponding subgroups of classical chains.

In 30, the same authors define and prove the existence of the decomposition in “connected components” for some objects which are very weak generalizations of surfaces (called flat chains). This result is used in the higher dimensional case of the previous paper to reduce the study of the jump set to the study of its “connected components” and then to understand their fine geometrical structure.

In 61, B. Merlet et al. study a variant of the Eikonal equation in 2D. The classical model arises in various physical models from the study of micromagnetism or liquid crystals and consists in the differential inclusion $\nabla u\in {𝕊}^1$ for functions $u:\Omega \subset {ℝ}^2\to ℝ$. The regularity and even the dimension of the singular set of $\nabla u$ assuming that its entropy production is a finite measure is still an important unsolved problem. Here, the authors consider instead the differential inclusion $\nabla u\in [-1,1]\times \left\{0\right\}\cup \left\{0\right\}\times [-1,1]$ or equivalently ${\partial }_{1}u{\partial }_{2}u=0$ (the circle is replaced by the cross formed by a vertical and a horizontal line). This model comes from anisotropic Ising models. In this variant, the regularity problem is easier and the authors show that the singular set is made of lines. They also establish similar results in higher dimension. This work introduces new lines of research for the study of the standard Eikonal equation.

In 28, motivated by some models of pattern formation involving an unoriented director field in the plane, B. Merlet, M. Pegon et al. study a family of unoriented counterparts to the Aviles–Giga functional. They introduce a nonlinear curl operator for such unoriented vector fields as well as a family of even entropies which they call "trigonometric entropies". Using these tools they show two main theorems which parallel some results in the literature on the classical Aviles–Giga energy. The first is a compactness result for sequences of configurations with uniformly bounded energies. The second is a complete characterization of zero-states, that is, the limit configurations when the energies go to 0. Their methods provide alternative proofs in the classical Aviles–Giga context.

In 19, J. Candau-Tilh, B. Merlet et al. consider a variant of the optimal transport problem and investigate the existence and characterization of its maximizers. In particular, they show that among sets of any fixed volume, the ball is the unique solution up to translation of this problem.

In 65, A. Natale studies a class of discrete models in which a collection of particles evolves in time following the gradient flow of an energy depending on the cell areas of an associated Laguerre (i.e. a weighted Voronoi) tessellation. Using a modulated energy argument, it is proven that in the limit of a high number of particles, the discrete solutions converge towards smooth solutions of nonlinear diffusion PDEs of porous medium type.

6.6 PhD theses

In his thesis entitled "Hybrid kinetic/fluid and structure preserving numerical methods for collisional kinetic equations", and defended on September 27 2024, T. Laidin focused on the development and analysis of efficient numerical methods for approximating solutions of potentially nonlinear kinetic collisional equations. These equations arise in various fields such as physics, notably in the study of semiconductors and gas dynamics. They also appear in biology in modelling the movement of cells within tissue. These models exhibit a multiscale aspect where there is, on one hand, a mesoscopic (or kinetic) description that gives the evolution of the distribution function of particles, molecules, or cells. On the other hand, through a process of averaging, one obtains the so-called macroscopic (or fluid) scale which allows to track the evolution of observable physical quantities: the moments of the distribution function. These moments correspond to the density, average velocity, and temperature of the considered particles. Throughout his manuscript, T. Laidin presents various ways to take advantage of fluid dynamics to construct and study efficient numerical methods for the kinetic scale. In the first part, he explores discretization methods aiming to preserve the structure of continuous equations. He begins by introducing an implicit finite volume scheme for a nonlinear reaction kinetic model. He studies the long-time behaviour of the discrete solution using hypocoercivity methods. Then, he examines a spectral method, based on general orthogonal polynomials, capable of preserving the moments of the solution while ensuring good convergence properties. The second part is dedicated to the design of numerical methods aiming to reduce the cost of kinetic simulations. To do this, he studies two approaches exploiting the evolution of the unknown’s moments. The first, a hybrid kinetic/fluid method, involves adopting dynamically and locally in position a less costly fluid description of the system instead of the more expensive kinetic one. The second approach also relies on the use of a fluid model, but this time to accelerate the temporal iterations of the method. Here, he proposes a prototype of a multiscale parareal method, using a fluid model as a coarse solver and a kinetic model as a fine solver.

In his thesis entitled "Theoretical and numerical analysis of perturbed isoperimetric problems", and defended on October 11 2024, J. Candau-Tilh focused on perturbed isoperimetric problems. These problems involve the minimisation of an energy composed of a perimeter term that promotes mass aggregation, countered by a perturbation term favouring disaggregation. In his manuscript, J. Candau-Tilh begins by presenting the concepts used as well as past and current research conducted on the isoperimetric problem and its variants. In Chapter 1, he studies a problem where the perimeter interacts with a non-local term called an exterior transport term, defined using optimal transport theory. He demonstrates the existence of solutions to this problem and, in regimes where the perimeter dominates, he proves that the minimisers are balls. Chapter 2 is dedicated to the exterior transport term. In a general framework, he shows that the variational problem defining it has solutions and a dual formulation. Using stronger assumptions, he finally shows that this term is maximised only by balls. In Chapter 3, he presents a numerical study in dimension 2 of the problem from Chapter 1. He approximates the minimisers of the energy considered via a gradient descent algorithm. The numerical results lead to conjecture the existence of a critical mass above which the minimisers are no longer balls, but elongated shapes with two axes of symmetry. Chapter 4 focuses on a perturbed isoperimetric problem where the perimeter and perturbation terms are not explicit. He exhibits a general set of assumptions under which a relaxed version of the problem admits minimisers. Under stronger hypotheses, he then investigates whether these minimisers have density estimates.

In his thesis entitled "Advanced numerical methods for high stiffness problems in reactive transport", realized in collaboration with IFPEN, and defended on November 18 2024, M. Jonval tackled the simulation of reactive transport problems in porous media, which is a major challenge for numerous IFPEN projects. Unfortunately, the performance of the reactive transport codes is nowadays strongly limited by numerical issues related to chemical modeling. These difficulties take various forms, all of which stemming from the stiffness of the equations to be solved. The goal of M. Jonval's thesis was to design new approaches for a more efficient resolution of the equations governing chemistry, first at equilibrium and then including kinetics. To this end, he made use of nonlinear preconditioning techniques. Besides, advanced time integration techniques adapted to stiffness have been investigated in order to by-pass difficulties coming from the intrinsic multi-scale aspect of chemical kinetics. Finally, the aforementioned contributions have been coupled with transport in porous media, with the ultimate goal of obtaining a numerical strategy which is robust with respect to large time steps.

7.1 Bilateral contracts with industry

C. Cancès 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). N.D.Q. 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.

The PhD thesis of M. Jonval (supervised by C. Cancès), that started in October 2021, was co-funded by INRIA (salaries) and IFPEN (overhead costs). The contract also followed the lines of the bilateral agreement between INRIA and IFPEN.

A 2-year research collaboration contract was signed in 2022 between INRIA and EDF R&D in the framework of the France Relance recovery plan. The contract followed the lines of the bilateral agreement between INRIA and EDF. The research project was coordinated by S. Lemaire and involved S. Pitassi, whose post-doc position (from 10/2022 to 07/2024) was funded in this framework ("Dispositif 4 de l'action de Préservation de l'Emploi de R&D"). The project concerned the development of high-order polyhedral methods for the numerical simulation of eddy current testing.

7.2 Bilateral grants with industry

In 2023, the PRCE project HIPOTHEC (HIgh-order POlyhedral meTHods for Eddy Current testing simulations) has been funded in the generic ANR call. This 6-year project, which started in January 2024, is coordinated by S. 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 8.2.2.

CEA (C. Bataillon) and ANDRA (L. Trenty) have been involved in the EURAD project on corrosion modeling together with the RAPSODI project-team (C. Cancès, C. Chainais-Hillairet, and B. Merlet). The final report is available 24.

8.1 International research visitors

8.2 National initiatives

8.3 Regional initiatives

9.1 Promoting scientific activities

9.2 Teaching - Supervision - Juries

9.3 Popularization

M. Alfaro has facilitated a workshop "Modèles d'invasions: des lapins de Fibonacci aux crapauds buffles" at the "Journées Nationales de l'APMEP (Association des Professeurs de Mathématiques de l'Enseignement Public)", in Le Havre, on October 21.

S. Lemaire is since September 2024 (co-) Scientific Officer in charge of Mediation for the INRIA Lille research center. He is responsible of both internal (organization of the monthly scientific event "30 MIN. de sciences") and external (through different actions, intended for high school students, in particular) scientific outreach. He co-organized on October 21-22 the "Rendez-vous des Jeunes Mathématiciennes et Informaticiennes" (RJMI), which is specifically geared towards female high school students, in order to promote scientific careers amongst them. In May, S. Lemaire also realized an intervention in the framework of the national scientific outreach program "1 scientifique, 1 classe / Chiche !" in a 2nde class of the Lycée Raymond Queneau in Villeneuve d'Ascq.

P. Gervais took part in the "Rendez-vous des Jeunes Mathématiciennes et Informaticiennes" (RJMI), which consisted on the first day, in a 3-hour research workshop on a simple geometric problem, and on the second day, in a 10-minute presentation.

J. Venel co-organized the internship Maths C pour L at Université Polytechnique Hauts-de-France. This event is open to around twenty female undergraduate mathematics students for one week. The students work in small groups on research topics and, thanks to various presentations by female researchers, doctoral students, and engineers, they can discover the various possible careers in mathematics. This week was co-funded by INRIA.

C. Chainais-Hillairet gave a talk at Lycée Faidherbe in Lille in order to promote the women's careers in mathematics. She was also in the organizing committee of the week-long event « Les Fourmis », proposed to secondary school girls experiencing gender non-mixing.

10.1 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 Computation89January 2020, 1093-1133HALDOIback 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 Analysis5542017, 1760--1785HALback to text
• 4 articleC.Clément Cancès, T.Thomas Gallouët and L.Léonard Monsaingeon. Incompressible immiscible multiphase flows in porous media: a variational approach.Analysis & PDE1082017, 1845--1876HALDOIback to text
• 5 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
• 6 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 Analysis404October 2020, 2473-2505HALDOIback to text
• 7 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 Analysis6142023, 1783--1818HALDOI
• 8 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 Sciences3212022, 175-207HALDOI
• 9 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, 461-472HALDOIback to text
• 10 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, 46-81HALDOIback to text
• 11 articleT.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
• 12 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

10.2 Publications of the year

10.3 Cited publications

• 67 articleA.Assyr Abdulle, M. E.Martin E. Huber and S.Simon Lemaire. An optimization-based numerical method for diffusion problems with sign-changing coefficients.Comptes Rendus. Mathématique35542017, 472--478DOIback to text
• 68 articleA.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 Analysis5242018, 1532-1567HALDOIback to text
• 69 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 Equations3612020, 133-153HALDOIback to text
• 70 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
• 71 inproceedingsM.Matthieu Alfaro and C.Claire Chainais-Hillairet. Finite Volume Scheme for~the~Diffusive Field-Road Model: Study of~the~Long Time Behaviour.Springer Proceedings in Mathematics & Statistics432Springer Proceedings in Mathematics & StatisticsStrasbourg, FranceSpringer Nature SwitzerlandOctober 2023, 215-223HALDOIback to text
• 72 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
• 73 articleR.R. Araya, C.C. Harder, D.D. Paredes and F.F. Valentin. Multiscale Hybrid-Mixed method.SIAM J. Numer. Anal.5162013, 3505--3531back to text
• 74 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-100002246DOIback to text
• 75 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/10.1007/978-3-319-70842-3DOIback to text
• 76 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, 333-340HALDOIback to text
• 77 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 Computation923402023, 635--693HALDOIback to text
• 78 articleR.Rafael Bailo and T.Thomas Rey. Projective and Telescopic Projective Integration for Non-Linear Kinetic Mixtures.Journal of Computational Physics45836 pages, 16 figures, 3 tablesJune 2022, 111082HALDOIback to text
• 79 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
• 80 inproceedingsS.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 IXBergen, NorwayJune 2020HALback to text
• 81 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. Acta55152010, 4451--4467back to textback to text
• 82 bookJ.J. Bear and Y.Y. Bachmat. Introduction to modeling of transport phenomena in porous media.4Springer1990back to text
• 83 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 HeritageVCSback to text
• 84 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)2312013, 199--214back to text
• 85 incollectionJ.-D.Jean-David Benamou, G.Guillaume Carlier and F.Filippo Santambrogio. Variational mean field games.Active Particles, Volume 1Springer2017, 141--171back to text
• 86 articleC.C. Besse. A relaxation scheme for the nonlinear Schrödinger equation.4232004, 934--952back to text
• 87 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 Mathematics253https://arxiv.org/abs/1601.00813September 2017, 147-168HALDOIback to text
• 88 articleM.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/10.1137/130913432DOIback to text
• 89 articleL.L. Boltzmann. Weitere Studien über das Wärmegleichgewicht unter Gasmolekl̎en.Wiener Berichte661872, 275--370back to text
• 90 articleF.F. Bouchut. Hypoelliptic regularity in kinetic equations..J. Math. Pures Appl. (9)81112002, 1135--1159DOIback to text
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