2023Activity 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  111, the multiphase and multicomponent character of the fluid which flows therein  76, and chemical reactions with a wide range of characteristic times  129. 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 70. We are interested in the devising of HHO methods in the curl/curl setting 108. 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 87, 116.

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 87 with a well suited mean-field self-interaction term 135. 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 and numerical simulation of multi-component systems

In 34, C. Cancès, C. Chainais-Hillairet et al. propose a two-point flux approximation finite-volume scheme for the approximation of two cross-diffusion systems coupled by a free interface to account for vapor deposition. The moving interface is addressed with a cut-cell approach, where the mesh is locally deformed around the interface. The scheme preserves the structure of the continuous system, namely: mass conservation, nonnegativity, volume-filling constraints and decay of the free energy. Numerical results illustrate the properties of the scheme.

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 45, 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.

In 46, C. Cancès, C. Chainais-Hillairet, B. Merlet, J. Venel et al. propose a new model to describe the evolution of the oxide layer covering iron in an aqueous environment. This model is an update of the original Diffusion Poisson Coupled Model (DPCM) proposed in 75 which is designed so that the Gibbs free energy is dissipated along time, making it consistent with the second principle of thermodynamics. The main changes with respect to the original model 75 are the use of nonlinear mobilities to represent the vacancy diffusion of iron cations and oxygen vacancies – these changes were already taken into account in the reduced model 16 –, and the fact that the oxide expansion is governed by the jump of grand canonical potential at the interfaces with the metal and the solution.

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 60, 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.

In 35, C. Cancès, M. Herda et al. present two finite volume approaches for modeling the diffusion of charged particles, specifically ions, in constrained geometries using a degenerate Poisson-Nernst-Planck system with cross-diffusion and volume filling. Both methods utilize a two-point flux approximation and are part of the exponentially fitted scheme framework. The only difference between the two is the selection of a Stolarsky mean for the drift term originating from a self-consistent electric potential. The first version of the scheme, referred to as (SQRA), uses a geometric mean and is an extension of the squareroot approximation scheme studied in 19. The second scheme, (SG), utilizes an inverse logarithmic mean to create a generalized version of the Scharfetter-Gummel scheme. Both approaches ensure the decay of some discrete free energy. Classical numerical analysis results - existence of discrete solutions, convergence of the scheme as the grid size and the time step go to 0 - follow. Numerical simulations show that both schemes are effective for moderately small Debye lengths, with the (SG) scheme demonstrating greater robustness in the small Debye length regime.

In 17, C. Cancès et al. propose a provably convergent Finite Volume scheme for the so-called Stefan–Maxwell model, which describes the evolution of the composition of a multi-component mixture and reads as a cross-diffusion system. The proposed scheme relies on a Two-Point Flux Approximation, and preserves at the discrete level some fundamental theoretical properties of the continuous model, namely the non-negativity of the solutions, the conservation of mass, and the preservation of the volume-filling constraints. In addition, the scheme satisfies a discrete entropy-entropy dissipation relation, very close to the relation which holds at the continuous level. In this article, C. Cancès et al. present the scheme together with its numerical analysis, and finally illustrate its behavior with some numerical results.

In 23, J. Moatti et al. consider a drift-diffusion model for light emitting diode simulations with high carrier densities. Different finite volume discretisations of the system are introduced, and the case of non-Boltzmann statistics is discussed. Numerical results illustrate the importance of using a consistent flux discretisation (as the SEDAN one) in order to avoid unphysical behaviours.

In 24, J. Moatti et al. study an aluminium gallium nitride ((Al,Ga)N) LED device thanks to numerical simulations. The computations highlight that the fluctuations induced by the ((Al,Ga)N) alloy increase the non-radiative Auger recombination in comparison to the radiative process, which could lead to a degradation of the device. Thus, this indicates that a careful design (using for example wider wells) is required to improve the efficiency of deep UV light emitters.

In 62, J. Moatti et al. consider an (In,Ga)N/GaN semiconductor system with quantum wells. They introduce a model with accounts for random alloy fluctuations through an atomistic tight-binding model, where quantum corrections are introduced via a localization landscape theory and propose a numerical method to simulate the model at hand. They investigate the impact of the order of the wells on the device behaviour and compare the results of the simulations with existing simulation codes as well as with physical experimentations.

In 18, C. Cancès et al. prove the existence of weak solutions to a system of two diffusion equations that are coupled by a pointwise volume constraint. The time evolution is given by gradient dynamics for a free energy functional. Their primary example is a model for the demixing of polymers, the corresponding energy is the one of Flory, Huggins and de Gennes. Due to the nonlocality in the equations, the dynamics considered here is qualitatively different from the one found in the formally related Cahn–Hilliard equations. Their angle of attack stems from the theory of optimal mass transport, that is, they consider the evolution equations for the two components as two gradient flows in the Wasserstein distance with one joint energy functional that has the volume constraint built in. The main difference with their previous work 97 is the nonlinearity of the energy density in the gradient part, which becomes singular at the interface between pure and mixed phases.

6.2 Analysis and numerical simulation in electromagnetism

In 50, T. Chaumont-Frelet et al. analyse Nédélec finite element discretizations of the time-harmonic Maxwell's equation, and show that the discrete solution is asymptotically optimal.

In 59, 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 51, 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.

Volume integral methods for the solution of eddy current problems are very appealing in practice since they require meshing only the conducting regions. However, they require the assembly and storage of a dense stiffness matrix. With the objective of cutting down assembly time and memory occupation, low-rank approximation techniques like the Adaptive Cross Approximation (ACA) have been considered a major breakthrough. Recently, the VINCO framework has been introduced to reduce significantly memory occupation and computational time thanks to a novel factorization of the dense stiffness matrix. In 32, S. Pitassi et al. introduce a new matrix compression technique enabled by the VINCO framework. They compare the performance of VINCO framework approaches with state-of-the-art alternatives in terms of memory occupation, computational time and accuracy by solving benchmark eddy current problems at increasing mesh sizes; the comparisons are carried out using both direct and iterative solvers. The results clearly indicate that the so-called VINCO-FAIME approach which exploits the Fast Multipole Method (FMM) has the best performance.

6.3 Structural properties of dissipative models and their discretization

In 43, 33, C. Chainais-Hillairet, M. Alfaro et al. consider the so-called field-road diffusion model in a bounded domain and its approximation by a TPFA finite volume scheme. In both the continuous and the discrete settings, they prove the exponential decay of an entropy, and thus the long time convergence to the stationary state selected by the total mass of the initial data. Numerical simulations confirm and complete the analysis, and raise new issues.

In 36, C. Chainais-Hillairet et al. define, in the case of quasilinear convection-diffusion equations, an approximation of the numerical fluxes obtained by extending the Scharfetter and Gummel fluxes. They show that this approximation is compatible with the asymptotic thermal equilibrium on an application example.

Following 36, in 20, 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 later 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 42, 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  63. 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 30, J. Moatti considers an anisotropic drift-diffusion system for semiconductor models in exterior magnetic fields. Following the methodology introduced in 104, he introduces an Hybrid Finite Volume scheme and proves the existence of discrete solutions with bounds on the densities, as well as a long-time behaviour result. Numerical experiments illustrate the behaviour of the scheme.

In 37, J. Moatti et al. present two different low-order structure-preserving schemes for drift-diffusion systems on general meshes. The two schemes have an entropic structure, which ensures the positivity of the computed densities, and are respectively based on the Discrete Duality Finite Volume and Hybrid Finite Volume methods. Numerical results assert the robustness of the schemes and show that their behaviours are similar.

In 39, J. Moatti introduces a new nonlinear high-order scheme for advection-diffusion. The scheme is based on the Hybrid High-Order methodology, and is devised in order to preserve the entropy structure of the continuous equation, which ensures the positivity of the solutions. Numerical experiments show that the scheme converges at the expected order, while preserving positivity. Moreover, numerical results point out that the use of high-order approximation gives a better efficiency (accuracy for a given computational cost) than low-order methods.

Following 39, in 28, 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 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 14, M. Herda et al. introduce and analyse numerical schemes for the homogeneous and the kinetic Lévy-Fokker-Planck equations. The discretizations are designed to preserve the main features of the continuous model such as conservation of mass, heavy-tailed equilibrium and (hypo)coercivity properties. They perform a thorough analysis of the numerical scheme and show exponential stability and convergence of the scheme. Along the way, they introduce new tools of discrete functional analysis, such as discrete nonlocal Poincaré and interpolation inequalities adapted to fractional diffusion. Their theoretical findings are illustrated and complemented with numerical simulations.

In 49, 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 58, 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 26, M. Herda et al. analyse a finite volume scheme for a nonlocal version of the Shigesada-Kawazaki-Teramoto (SKT) cross-diffusion system. They prove the existence of solutions to the scheme, derive qualitative properties of the solutions and prove its convergence. The proofs rely on a discrete entropy-dissipation inequality, discrete compactness arguments, and on the novel adaptation of the so-called duality method at the discrete level. Finally, thanks to numerical experiments, they investigate the influence of the nonlocality in the system: on convergence properties of the scheme, as an approximation of the local system and on the development of diffusive instabilities.

In 15, C. Calgaro, C. Cancès and E. Creusé performed 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  89 emanating from the team. The proof proposed in 15 relies on compactness arguments. In order to treat the Joule effect term, a second order discrete Gagliardo-Nirenberg inequality has been 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 Assessment and improvement of the efficiency of numerical methods

In 44, C. Cancès, M. Jonval et al. propose two numerical strategies to solve chemical equilibria in aqueous solutions. The first strategy can be thought as an extension of the parametrisation trick initially proposed in 85 to smoothen the stiff relation between molar fractions and chemical potential. In a second approach referred to as Cartesian representation, this relation is relaxed and only recovered at convergence. Numerical experiments show that our approaches allow to strongly improve the robustness of the Newton-Raphson algorithm with respect to state-of-the-art methods even without the adjunction of limitation strategies.

In 38, T. Laidin and T. Rey present an extension of the hybrid, model-adaptation method introduced in 27 for linear collisional kinetic equations in a diffusive scaling to the nonlinear mean-field Vlasov-Poisson-BGK model. The aim of the approach is to reduce the computational cost by taking advantage of the lower dimensionality of the asymptotic model. The authors show that the method offers a significant computational gain for the nonlinear model and illustrate properties such as mass conservation and long-time behaviour of the hybrid model.

In 21, E. Creusé et al. present a unified framework for goal-oriented estimates for elliptic and parabolic problems that combine the dual-weighted residual method with equilibrated flux reconstruction. This framework allows to analyze simultaneously different approximation schemes for the space discretization of the primal and the dual problems, such as conforming or nonconforming finite element methods, discontinuous Galerkin methods, or finite volume methods. Their main contribution is the splitting of the error on the quantity of interest into a fully computable estimator and a remainder that is bounded, up to an explicit constant, by the product of the fully computable estimators of the primal and dual problems. Some illustrative numerical examples that validate their theoretical results are presented.

6.5 Analysis and numerical simulation of variational models

In 47, 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 57, B. Merlet 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 53 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 56, 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 54, 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 crossed 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 55, 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 standard perimeter for Caccioppoli sets and $P_{\epsilon }$ is a nonlocal energy converging to the perimeter as $\epsilon$ vanishes. Their 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 25, 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 48, 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 61, 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.

In 52, A. Natale et al. introduce a time discretization for Wasserstein gradient flows based on the classical Backward Differentiation Formula of order two. The main building block of the scheme is the notion of geodesic extrapolation in the Wasserstein space, which in general is not uniquely defined. They propose several possible definitions for such an operation, and prove convergence of the resulting scheme to the limit PDE, in the case of the Fokker-Planck equation. For a specific choice of extrapolation they also prove a more general result, that is convergence towards EVI flows. Finally, they propose a variational finite volume discretization of the scheme which numerically achieves second order accuracy in both space and time.

In 22, E. Facca, A. Natale et al. address the numerical solution of the quadratic optimal transport problem in its dynamical form, the so-called Benamou-Brenier formulation. When solved using interior point methods, the main computational bottleneck is the solution of large saddle-point linear systems arising from the associated Newton-Raphson scheme. They compare different preconditioners to solve these linear systems via iterative methods, and they introduce a new one based on the partial commutation of the operators that compose the dual Schur complement of these saddle-point linear systems, which scales only slightly more than linearly with respect to the number of unknowns used to discretize the problem.

In 29, E. Facca et al. study the emergence of loop in complex networks in optimal transport problems with time-varying loads. The study includes both theoretical and experimental results, with a test-case defined on the Bordeaux bus network.

7.1 Bilateral contracts with industry

A research collaboration contract has been signed in 2022 between RAPSODI and EDF R&D in the framework of the France Relance recovery plan. The contract follows the lines of the bilateral agreement between Inria and EDF. The research project is coordinated by S. Lemaire and involves S. Pitassi, whose 2-year post-doc position, which started in October 2022, has been funded in this framework ("Dispositif 4 de l'action de Préservation de l'Emploi de R&D"). The project concerns the development of high-order polyhedral methods for the numerical simulation of eddy current testing.

In 2023, the PRCE project HIPOTHEC (HIgh-order POlyhedral meTHods for Eddy Current testing simulations) has been funded in the generic ANR call. This project 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 RAPSODI and EDF R&D. More details about the HIPOTHEC project can be found in Section 8.4.1.

The PhD thesis of M. Jonval (supervised by C. Cancès), that started in October 2021, is co-funded by Inria (salaries) and IFPEn (overhead costs). The contract follows the lines of the bilateral contract between Inria and IFPEn.

7.2 Bilateral grants with industry

CEA (Christian Bataillon) and ANDRA (Laurent Trenty) are involved in the EURAD project on corrosion modeling together with the RAPSODI project-team (C. Cancès, C. Chainais-Hillairet, and B. Merlet). More details on the project can be found in Section 8.3.1.

8.1 International initiatives

8.2 International research visitors

8.3 European initiatives

8.4 National initiatives

8.5 Regional initiatives

9.1 Promoting scientific activities

9.2 Teaching - Supervision - Juries

9.3 Popularization

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

• 63 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
• 64 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
• 65 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
• 66 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
• 67 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
• 68 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
• 69 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
• 70 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
• 71 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
• 72 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
• 73 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
• 74 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
• 75 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 textback to textback to text
• 76 bookJ.J. Bear and Y.Y. Bachmat. Introduction to modeling of transport phenomena in porous media.4Springer1990back to text
• 77 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
• 78 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
• 79 incollectionJ.-D.Jean-David Benamou, G.Guillaume Carlier and F.Filippo Santambrogio. Variational mean field games.Active Particles, Volume 1Springer2017, 141--171back to text
• 80 articleC.C. Besse. A relaxation scheme for the nonlinear Schrödinger equation.4232004, 934--952back to text
• 81 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
• 82 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
• 83 articleL.L. Boltzmann. Weitere Studien über das Wärmegleichgewicht unter Gasmolekl̎en.Wiener Berichte661872, 275--370back to text
• 84 articleY.Yann Brenier. Décomposition polaire et réarrangement monotone des champs de vecteurs.CR Acad. Sci. Paris Sér. I Math.3051987, 805--808back to text
• 85 articleK.K. Brenner and C.C. Cancès. Improving Newton's Method Performance by Parametrization: The Case of the Richards Equation.SIAM J. Numer. Anal.5542017, 1760-1785back to text
• 86 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.162016, 123--139back to text
• 87 articleY.Yunhai Cai. Linear theory of microwave instability in electron storage rings.Physical Review Special Topics-Accelerators and Beams1462011, 061002back to textback to text
• 88 articleJ.J. Cai and J.J. Shen. Two classes of linearly implicit local energy-preserving approach for general multi-symplectic Hamiltonian PDEs.4012020, 108975back to text
• 89 articleC.C. Calgaro, C.C. Colin and E.E. Creusé. A combined finite volume-finite element scheme for a low-Mach system involving a Joule term.AIMS Math.512020, 311--331URL: https://doi.org/10.3934/math.2020021DOIback to text
• 90 articleC.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 Analysis4112021, 271-314HALDOIback to text
• 91 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 Equations352https://arxiv.org/abs/1801.094082019, 545-575HALDOIback to text
• 92 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, 2544-2571HALDOIback to text
• 93 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
• 94 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, 1123-1152HALDOIback to text
• 95 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
• 96 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/2001.112222020, pp. 2784-2710HALDOIback to text
• 97 articleC.C. Cancès, D.D. Matthes and F.F. Nabet. A two-phase two-fluxes degenerate Cahn--Hilliard model as constrained Wasserstein gradient flow.Arch. Ration. Mech. Anal.23322019, 837--866back to text
• 98 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--1003HALDOIback to text
• 99 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 Computation903282021, 517-563HALDOIback to text
• 100 articleC.Clément Cancès and A.Antoine Zurek. A convergent finite volume scheme for dissipation driven models with volume filling constraint.Numerische Mathematik1512022, 279--328HALDOIback to textback to text
• 101 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, 1-36HALDOIback to text
• 102 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.13312001, 1--82URL: http://dx.doi.org/10.1007/s006050170032DOIback to text
• 103 articleC.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/drl045DOIback to text
• 104 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, 963-1016HALDOIback to textback to textback to text
• 105 articleC.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 Numérique5012016, 135-162HALback to text
• 106 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 Analysis5222018, 457-480HALDOIback to text
• 107 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 Analysis5612022, 261-285HALDOIback to text
• 108 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, 175-207HALDOIback to text
• 109 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, 723-748HALDOIback to text
• 110 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
• 111 bookO.O. Coussy. Poromechanics.John Wiley & Sons2004back to text
• 112 inproceedingsM.Marco Cuturi. Sinkhorn Distances: Lightspeed Computation of Optimal Transport.Proceedings of the 26th International Conference on Neural Information Processing Systems - Volume 2NIPS'13Red Hook, NY, USALake Tahoe, NevadaCurran Associates Inc.2013, 2292–2300back to text
• 113 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.2832015, 1--21back to text
• 114 articleA.A. Dragulescu and V. M.V. M. Yakovenko. Statistical mechanics of money.The European Physical Journal B-Condensed Matter and Complex Systems1742000, 723--729back to text
• 115 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
• 116 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--639back to text
• 117 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--296back to text
• 118 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 Analysis5432022HALDOIback to text
• 119 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 EngineeringChamSpringer International Publishing2019, 45--60DOIback to text
• 120 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, 99-149HALDOIback to text
• 121 articleC. ..C .W. Gear and I. ..I .G. Kevrekidis. Projective methods for stiff differential equations: problems with gaps in their eigenvalue spectrum.SIAM J. Sci. Comput.2442003, 1091--1106back to text
• 122 articleS.S. Gottlieb, C.-W.C.-W. Shu and E.E. Tadmor. Strong stability-preserving high-order time discretization methods.SIAM Rev.4312001, 89--112URL: https://doi.org/10.1137/S003614450036757Xback to text
• 123 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 123https://arxiv.org/abs/1610.051382019, 593-636HALDOIback to text
• 124 articleP.-E.Pierre-Emmanuel Jabin and T.Thomas Rey. Hydrodynamic limit of granular gases to pressureless Euler in dimension 1.Quarterly of Applied Mathematics75https://arxiv.org/abs/1602.09103 - 26 pages, 1 figure2017, 155-179HALDOIback to text
• 125 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
• 126 articleL.L. Kantorovitch. On the translocation of masses.C. R. (Dokl.) Acad. Sci. URSS, n. Ser.371942, 199--201back to text
• 127 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 B3412013, 113--138back to text
• 128 articleP.P. Lafitte and G.G. Samaey. Asymptotic-preserving Projective Integration Schemes for Kinetic Equations in the Diffusion Limit.3422012, A579–-A602back to textback to text
• 129 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
• 130 articleS.Simon Lemaire. Bridging the hybrid high-order and virtual element methods.IMA J. Numer. Anal.4112021, 549--593DOIback to text
• 131 articleD.D. Matthes and S.S. Plazzotta. A variational formulation of the BDF2 method for metric gradient flows.ESAIM: M2AN5312019, 145-172back to text
• 132 inproceedingsW.Ward Melis, T.Thomas Rey and G.Giovanni Samaey. Projective integration for nonlinear BGK kinetic equations.Finite Volumes for Complex Applications VIII200Hyperbolic, Elliptic and Parabolic ProblemsProceedings FVCA 8Lille, FranceSpringer International PublishingJune 2017, 155-162HALDOIback to text
• 133 articleA.A. Mielke. A gradient structure for reaction-diffusion systems and for energy-drift-diffusion systems.Nonlinearity2442011, 1329--1346URL: http://dx.doi.org/10.1088/0951-7715/24/4/016DOIback to text
• 134 softwareversionA.Alexandre Mouton and T.Thomas Rey. KINEBEC - Numerical simulation of Boltzmann-Norheim equation.1.0December 2021 lic: GNU General Public License v3.0 only.HALSoftware Heritageback to text
• 135 articleJ.JB Murphy, R.RL Gluckstern and S.S Krinsky. Longitudinal wake field for an electron moving on a circular orbit.Part. Accel.57BNL-630901996, 9--64back to text
• 136 articleA.Andrea Natale and G.Gabriele Todeschi. A mixed finite element discretization of dynamical optimal transport.Journal of Scientific Computing9122022, 1--26back to text
• 137 articleA.Andrea Natale and G.Gabriele Todeschi. Computation of optimal transport with finite volumes.ESAIM: Mathematical Modelling and Numerical Analysis555September 2021, 1847-1871HALDOIback to textback to text
• 138 inproceedingsA.Andrea Natale and G.Gabriele Todeschi. TPFA finite volume approximation of Wasserstein gradient flows.International Conference on Finite Volumes for Complex ApplicationsSpringer2020, 193--201back to text
• 139 articleL.L. Onsager. Reciprocal relations in irreversible processes. II..Physical Review381931, 2265--2279back to text
• 140 articleL.Lorenzo Pareschi and T.Thomas Rey. Moment preserving Fourier-Galerkin spectral methods and application to the Boltzmann equation.SIAM Journal on Numerical Analysis606December 2022, 3216-3240HALDOIback to text
• 141 articleL.Lorenzo Pareschi and T.Thomas Rey. On the stability of equilibrium preserving spectral methods for the homogeneous Boltzmann equation.Applied Mathematics Letters120https://arxiv.org/abs/2011.05811November 2021, 107187HALback to text
• 142 articleM. A.Mark A. Peletier and M.Matthias Röger. Partial Localization, Lipid Bilayers, and the Elastica Functional.1933September 2009, 475--537DOIback to text
• 143 unpublishedM. A.M. A. Peletier. Variational Modelling: Energies, gradient flows, and large deviations.2014, arXiv:1402.1990back to text
• 144 articleG.Gabriel Peyré, M.Marco Cuturi and others. Computational optimal transport.Center for Research in Economics and Statistics Working Papers2017-862017back to text
• 145 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 2014HALback to text
• 146 bookF.Filippo Santambrogio. Optimal transport for applied mathematicians.Birkäuser, NY5558-63Springer2015, 94back to text
• 147 articleJ.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
• 148 articleB.B. Seguin and N. J.N. J. Walkington. Multi-Component Multiphase Flow Through a Poroelastic Medium.J. Elasticity1352019, 485-507back to text
• 149 articleB.B. Seguin and N. J.N. J. Walkington. Multi-component Multiphase Porous Flow.Arch. Ration. Mech. Anal.2352020, 2171-2196URL: https://doi.org/10.1007/s00205-019-01473-7back to text
• 150 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--100back to text
• 151 unpublishedF.F. Sma\"i. A thermodynamic formulation for multiphase compositional flows in porous media.2020, HAL: hal-02925433back to text