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 assessed numerical methods. The 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 enjoy strong theoretical foundations allowing to guarantee their behavior in very general situations. They are also 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 particles systems, the curse of dimensionality makes numerical computations realistically feasible only if specific computationally efficient numerical strategies are used.

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 assessed 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  91, the multiphase and multicomponent character of the fluid which flows therein  61, and chemical reactions with a wide range of characteristic times  110. 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 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 57. We are interested in the devising of HHO methods in the curl/curl setting 22. 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.

M. Herda and A. Natale 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 epididemic 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 71, 96.

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 71 with a well suited mean-field self-interaction term 117. 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 Platform ParaSkel++

Family={research, vehicle}; Audience={partners}; Evolution={lts}; Duration={2}; Contribution={leader}; URL={gitlab.inria.fr/simlemai/paraskel}

ParaSkel++62 is a C++ platform, conceived by S. Lemaire and mainly developed since December 2022 by T. Zoto (succeeding L. Beaude), which is freely distributed under LGPL v3.0. The ParaSkel++ platform aims at the high-performance, arbitrary-order, 2/3D numerical approximation of PDEs on general polytopal meshes using skeletal Galerkin methods. Skeletal Galerkin methods form a vast family of numerical methods for the approximation of PDE-based models that satisfy the following two building principles (see 111):

• $•$
  the degrees of freedom (DOF) of the method split into (i) skeleton DOF, attached to the geometric entities (vertices, edges, faces) composing the mesh skeleton and common to all cells sharing the geometric entity in question, which prescribe the conformity properties of the underlying discrete functional space, and (ii) bulk DOF (if need be), attached to the interior of the cells, which play no role in the prescription of the conformity properties of the underlying discrete functional space;
• $•$
  the global discrete bilinear form of the problem (potentially after linearization, if the problem is nonlinear) writes as the sum over the mesh cells of cell-wise (referred to as local) bilinear contributions.

The very structure underpinning skeletal methods grants them the property of being amenable to static condensation, i.e. locally to each cell, bulk DOF can be eliminated in terms of the local skeleton DOF by means of a Schur complement. The final global system to solve thus writes in terms of the skeleton DOF only. The skeletal family encompasses in particular standard FE methods and virtual-like Galerkin methods (VEM, HDG, HHO...). It does not contain (plain vanilla) DG methods. ParaSkel++ offers a high-performance factorized C++ architecture for the implementation of arbitrary-order skeletal methods on general 2/3D polytopal meshes. A first version (v1, August 2021) of the platform is operational, featuring a sequential implementation of all the main skeletal methods.

5.2 Code KINEBEC

Family={vehicle}; Audience={partners}; Evolution={lts}; Duration={2}; Contribution={leader};

URL={Software Heritage deposit}

KINEBEC (Kinetic Bose–Einstein Condensates)  116 is a C code (freely distributed under GPL), developed by Alexandre Mouton (CNRS permanent engineer at Université de Lille) and T. Rey, devoted to the simulation of collisional kinetic equations of Boltzmann type using a deterministic, spectral Galerkin approach. While mainly devoted to the numerical simulation of the Boltzmann–Nordheim equation (BNE) for fermions and bosons, this code can also be used to solve the classical Boltzmann equation (BE). It relies on state-of-the-art fast spectral approaches to solve with high accuracy and efficiency both the BNE and BE. It has been parallelized on shared memory (OpenMP), but also has a native MPI support for heterogeneous architectures, as well as CUDA capabilities.

5.3 Other codes

6.1 Modeling and numerical simulation of multi-component systems

The mathematical and numerical modeling of corrosion is a longstanding topic shared by several members of the RAPSODI research team. In the framework of the H2020 European program EURAD on the management of nuclear wastes, C. Cancès, C. Chainais-Hillairet, B. Merlet, F. Raimondi and J. Venel proposed in 32 a modification of a reduced model for the corrosion of the iron to make it compatible with the second principle of the thermodynamics, in the sense that the free energy is decaying along time. This remarkable stability property and uniform bounds derived thanks to the Moser iteration technique allow to rigorously establish the global in time existence of solutions to the model. In particular, and as highlighted by numerical results presented in 32, the new model preserves some physical bounds that are sometimes overpassed by the preceding models of the literature. The presence of moving interfaces is a characteristic 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 45 where 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.

One of the key modifications proposed in 32 to make the model compatible with thermodynamics is the fact that the evolution of the iron cations follows a vacancy diffusion process, leading to nonlinear convection, while diffusion remains linear. As a consequence, Scharfetter–Gummel fluxes can no longer be used to compute a finite volume approximation of the iron cation concentration. In 18, C. Cancès and J. Venel proposed a new finite volume scheme to compute approximate solutions to such nonlinear convection diffusion equations. The scheme, which extends to the nonlinear mobility setting the so-called square-root approximation finite volume scheme  112, enjoys several remarkable properties, among which the decay at the discrete level of the free energy. By properly quantifying the dissipation of the free energy, C. Cancès and J. Venel show the convergence of the scheme when the discretization parameters (mesh size and time step) tend to 0 in the case of a scalar nonlinear convection diffusion equation. Numerical experiments show that the scheme is very efficient, justifying its use in the more complex setting of 32.

In 17, 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 81 is the nonlinearity of the energy density in the gradient part, which becomes singular at the interface between pure and mixed phases.

In 15, T.Rey et al. study a stochastic individual-based model of interacting plant and pollinator species through a bipartite graph: each species is a node of the graph, an edge representing interactions between a pair of species. The dynamics of the system depends on the between- and within-species interactions: pollination by insects increases plant reproduction rate but has a cost which can increase plant death rate, depending on pollinators density. Pollinators reproduction is increased by the resources harvested on plants. Each species is characterized by a trait corresponding to its degree of generalism. This trait determines the structure of the interactions graph and the quantity of resources exchanged between species. Their model includes in particular nested or modular networks. Deterministic approximations of the stochastic measure-valued process by systems of ordinary differential equations or integro-differential equations are established and studied as in 99, when the population is large or when the graph is dense and can be replaced with a graphon. The long-time behaviors of these limits are studied and central limit theorems are established to quantify the difference between the discrete stochastic individual-based model and the deterministic approximations. Finally, studying the continuous limits of the interaction network and the resulting PDEs, they show that nested plant-pollinator communities are expected to collapse towards a coexistence between a single pair of species of plants and pollinators.

In 48, 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.

6.2 Analysis and numerical simulation in electromagnetism

In 23, E. Creusé et al. study the $𝐀-\phi -𝐁$ magnetodynamic Maxwell system, given in its potential and space-time formulation. First, the existence of strong solutions with the help of the theory of Showalter on degenerate parabolic problems is established. Second, using energy estimates, the existence and the uniqueness of the weak solution to the $𝐀-\phi -𝐁$ system is inferred.

In 22, S. Lemaire et al. prove a discrete version of the first Weber inequality on three-dimensional hybrid spaces spanned by vectors of polynomials attached to the elements and faces of a polyhedral mesh. They then introduce two Hybrid High-Order methods for the approximation of the magnetostatics model, in both its (first-order) field and (second-order) vector potential formulations. These methods are applicable on general polyhedral meshes, and allow for arbitrary orders of approximation. Leveraging the previously established discrete Weber inequality, they perform a comprehensive analysis of the two methods, that they finally validate on a set of test-cases.

6.3 Structural properties of dissipative models and their discretization

In 30 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 of 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 their theoretically proven properties numerically, simulate a realistic device setup and show exponential decay in time with respect to the $L^2$ error as well as a physically and analytically meaningful relative entropy.

In 12 M. Herda et al. introduce and analyse numerical schemes for the homogeneous and the kinetic Lévy-Fokker-Planck equation. 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 19, C. Cancès et al. propose and study an implicit finite volume scheme for a general model which describes the evolution of the composition of a multi-component mixture in a bounded domain. They assume that the whole domain is occupied by the different phases of the mixture, which leads to a volume filling constraint. In the continuous model, this constraint yields the introduction of a pressure, which should be thought as a Lagrange multiplier for the volume filling constraint. The pressure solves an elliptic equation, to be coupled with parabolic equations, possibly including cross-diffusion terms, which govern the evolution of the mixture composition. The system admits an entropy structure, which is the cornerstone of the analysis. The main objective of their work is the design of a two-point flux approximation finite volume scheme which preserves the key properties of the continuous model, namely the volume filling constraint and the control of the entropy production. Thanks to these properties, and in particular to the discrete entropy-entropy dissipation relation, the authors are able to prove the existence of solutions to the scheme and its convergence. Finally, they illustrate the behavior of their scheme through different applications.

In 13, T. Rey et al. propose fully explicit projective integration and telescopic projective integration schemes for the multispecies Boltzmann and BGK equations. Projective integration has been recently proposed as a viable alternative to fully implicit and micro-macro methods for providing light, nonintrusive and almost AP integrators for collisional kinetic equations. The methods employ a sequence of small forward-Euler steps, intercalated with large extrapolation steps. The telescopic approach repeats said extrapolations as the basis for an even larger step. This hierarchy renders the computational complexity of the method essentially independent of the stiffness of the problem, which permits the efficient solution of equations in the hyperbolic scaling with very small Knudsen numbers. The schemes are validated on a range of scenarios, demonstrating their prowess in dealing with extreme mass ratios, fluid instabilities, and other complex phenomena.

In 35, C. Chainais-Hillairet et al. propose a new numerical 2-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 relative entropy decay properties.

In 20, C. Chainais-Hillairet, M. Herda, S. Lemaire and J. Moatti devise and study three Hybrid Finite Volume methods for an heterogeneous and anisotropic linear advection-diffusion equation on general meshes. They consider two linear methods, as well as a new, nonlinear scheme. They prove the existence of a solution to each scheme, and positivity of the discrete solutions to the nonlinear scheme. For the three schemes, they show that the discrete solutions converge exponentially fast in time towards their associated discrete steady-states. Their theoretical results are illustrated by numerical simulations.

In 46, J. Moatti considers an anisotropic drift-diffusion system for semiconductors models in exterior magnetic fields. Following the methodology introduced in 20, 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 43 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 29, F. Raimondi tackles the homogenization of a quasilinear elliptic problem having a singular lower-order term and posed in a two-component domain with an $ϵ$-periodic imperfect interface. A Dirichlet condition is prescribed on the exterior boundary, while the continuous heat flux is assumed to be proportional to the jump of the solution on the interface via a function of order ${ϵ}^{\gamma }$. An homogenization result for $-1<\gamma <1$ is proved by means of the periodic unfolding method, adapted to two-component domains by P. Donato, K. H. Le Nguyen and R. Tardieu. One of the main tools in the homogenization process is the study of a suitable auxiliary linear problem and a related convergence result. It shows that the gradient of $u^{ϵ}$ behaves like that of the solution of the auxiliary one, associated with a weak cluster point of the sequence $\left\{u^{ϵ}\right\}$, as $ϵ\to 0$. This allows not only to pass to the limit in the quasilinear term, but also to study the singular term near its singularity, via an accurate a priori estimate.

In 33, 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.

6.4 Assessment and improvement of the efficiency of numerical methods

In 44, T. Laidin presents a hybrid numerical method that allows to reduce the computational cost of linear collisional kinetic equations. The method relies on an accurate dynamic domain decomposition technique. In addition to a rigorous study of the conservation of mass, he shows that the method is very efficient. Several numerical experiments also illustrate the properties of the method such as the conservation of mass and the long-time behaviour of discrete solutions.

In 36, E. Creusé et al. present unified frameworks for goal-oriented estimates for elliptic and parabolic problems that combine the dual-weighted residual method with equilibrated flux reconstruction. These frameworks allow to analyze simultaneously different approximation schemes for the space discretization of the primal and the dual problems, as conforming or nonconforming finite element method, discontinuous Galerkin methods, or finite volume method. 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.

In 31, 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  73 emanating from the team. The proof proposed in 31 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.

In 21, S. Lemaire et al. establish the equivalence between the Multiscale Hybrid-Mixed (MHM) 55 and the Multiscale Hybrid High-Order (MsHHO) 89 methods for a variable diffusion problem with piecewise polynomial source term. Under the idealized assumption that the local problems defining the multiscale basis functions are exactly solved, they prove that the equivalence holds for general polytopal (coarse) meshes and arbitrary approximation orders. They finally leverage the interchange of properties to perform a unified convergence analysis, as well as to improve on both methods.

In 47, T. Rey et al. present an efficient implementation of a spectral Fourier-Galerkin algorithm for the quantum Boltzmann–Nordheim equation (BNE) for fermions and bosons. The BNE was first formulated by Uehling and Uhlenbeck starting from a classical Boltzmann equation with heuristic arguments. Using novel parallelization techniques, they investigate some of the conjectured properties of the large time behavior of the solutions to this equation. In particular, they are able to observe numerically both Bose–Einstein condensation and Fermi–Dirac relaxation, and to make some conjectures on their stability.

In 14, S. Bassetto, C. Cancès et al. benchmark several numerical approaches building on upstream mobility two-point flux approximation finite volumes to solve Richards' equation in domains made of several rocktypes. Their study encompasses four different schemes corresponding to different ways to approximate the nonlinear transmission condition systems arising at the interface between different rocks, as well as different resolution strategies based on Newton's method with variable switch. The different methods are compared on filling and drainage test-cases with standard nonlinearities of Brooks–Corey and van Genuchten type, as well as with challenging steep nonlinearities.

In 37 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. The main purpose of this paper is to design efficient preconditioners to solve these linear systems via iterative methods. Among the proposed preconditioners, they introduce one based on the partial commutation of the operators that compose the dual Schur complement of these saddle point linear systems, which they refer as $ℬℬ$-preconditioner. A series of numerical tests show that the $ℬℬ$-preconditioner is the most efficient among those presented, with a CPU-time scaling only slightly more than linearly with respect to the number of unknowns used to discretize the problem.

In 28, T. Rey et al. introduce a novel Fourier-Galerkin spectral method that improves the classical spectral method by making it conservative on the moments of the approximated distribution, without sacrificing its spectral accuracy or the possibility of using fast algorithms. The method is derived directly using a constrained best approximation in the space of trigonometric polynomials and can be applied to a wide class of problems where preservation of moments is essential. The authors then apply the new spectral method to the evaluation of the Boltzmann collision term, and prove spectral consistency and stability of the resulting Fourier-Galerkin approximation scheme. They illustrate their theoretical findings by various 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 49 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.

6.5 Analysis and numerical simulation of variational models

In 34, 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, convergence of 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 26, A. Natale et al. develop a novel particle discretization for compressible isentropic fluids and porous media flow. The main idea of the method is to replace the internal energy of the fluid by its Moreau–Yosida regularization in the $L^2$ sense, which can be efficiently computed as a semi-discrete optimal transport problem. Using a modulated energy argument which exploits the convexity of the energy in Eulerian variables, they prove quantitative convergence estimates towards smooth solutions.

In 38, 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 24, E. Facca et al. give a new characterization of the cut locus of a point on a compact Riemannian manifold as the zero set of the optimal transport density solution of the Monge–Kantorovich equations, a PDE formulation of the optimal transport problem with cost equal to the geodesic distance. Combining this result with an optimal transport numerical solver based on the so-called dynamical Monge–Kantorovich approach, they propose a novel framework for the numerical approximation of the cut locus of a point in a manifold. They show the applicability of the proposed method on a few examples settled on 2d-surfaces embedded in ${ℝ}^3$ and discuss advantages and limitations.

In wet-lab experiments, the slime mold Physarum polycephalum has demonstrated its ability to tackle a variety of computing tasks, among them the computation of shortest paths and the design of efficient networks. For the shortest path problem, a mathematical model for the evolution of the slime is available and it has been shown in computer experiments and through mathematical analysis that the dynamics solves the shortest path problem. In 16, E. Facca et al. generalize the dynamics to the network design problem. They formulate network design as the problem of constructing a network that efficiently supports a multi-commodity flow problem. They investigate the dynamics in computer simulations and analytically. The simulations show that the dynamics is able to construct efficient and elegant networks. In the theoretical part they show that the dynamics minimizes an objective combining the cost of the network and the cost of routing the demands through the network. They also give alternative characterizations of the optimum solution.

In25 E. Facca et al. introduce the transport energy functional $ℰ$ (a variant of the Bouchitté-Buttazzo-Seppecher shape optimization functional) and they prove that its unique minimizer is the optimal transport density ${\mu }^{*}$, i.e., the solution of Monge-Kantorovich partial differential equations. They study the gradient flow of $ℰ$ and show that ${\mu }^{*}$ is the unique global attractor of the flow. Next they introduce a two parameter family ${\left\{{ℰ}_{\lambda ,\delta }\right\}}_{\lambda ,\delta >0}$ of strictly convex regularized functionals approximating $ℰ$ and they prove the convergence of the minimizers ${\mu }_{\lambda ,\delta }^{*}$ of ${ℰ}_{\lambda ,\delta }$ to ${\mu }^{*}$ as they let $\delta \to 0^{+}$ and $\lambda \to 0^{+}.$ They derive an evolution system of fully non-linear PDEs as the gradient flow of ${ℰ}_{\lambda ,\delta }$ in $L^2$, showing existence and uniqueness of the solution for all times. They are able to prove that the trajectories of the flow converge in $W_0^{1,p}$ to the unique minimizer ${\mu }_{\lambda ,\delta }^{*}$ of ${ℰ}_{\lambda ,\delta }.$ This allows them to characterize ${\mu }_{\lambda ,\delta }^{*}$ by a non-linear system of PDEs which turns out to be a perturbation of Monge-Kantorovich equations by a p-Laplacian.

B. Merlet et al. continued their work about the characterization of objects which split as cartesian products. Their previous work leads to the study of generalized surfaces of dimension $k=k_1+k_2$ in ${ℝ}^n={ℝ}^{n_1}\times {ℝ}^{n_2}$ which write as cartesian products ${\Sigma }_1\times {\Sigma }_2$ with ${\Sigma }_1\subset {ℝ}^{n_1}$ of dimension $k_1$ and ${\Sigma }_2\subset {ℝ}^{n_2}$ of dimension $k_2$. The generalized surface they consider are the so called normal rectifiable $k$-chains. In 42 they establish a rigidity result for such objects, namely given some normal rectifiable $k$-chain $A$, they show that if at almost every $x$, the tangent $k$-plane to $A$ at $x$ splits as a cartesian product $L^1\left(x\right)\times L^2\left(x\right)$ with $L^1\left(x\right)\subset {ℝ}^{n_1}$ of dimension $k_1$ and $L^2\left(x\right)\subset {ℝ}^{n_2}$ of dimension $k_2$ then $A$ is supported in a countable union of cartesian products of the form described above: ${\cup }_j{\Sigma }_1^j\times {\Sigma }_2^j$.

As a preliminary work, B. Merlet et al. establish in 41 that normal rectifiable chains admit a decomposition in a (at most) countable number of connected components.

In 42 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 39 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 40, 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 27, 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.

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 ANR France Relance funding plan. The contract follows the lines of the bilateral agreement between Inria and EDF. The collaboration involves S. Lemaire and S. Pitassi, and concerns the development of high-order polyhedral methods for the numerical simulation of eddy current testing. The 2-year post-doc position of S. Pitassi, which started in October 2022, is funded by this grant ("Dispositif 4"). It has also been agreed that the ParaSkel++ code will be the prototyping platform for this project.

The PhD thesis of M. Jonval, 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, B. Merlet, and F. Raimondi). More details on the project can be found in Section 8.3.1.

In the framework of the MSCA's DATAHYKING project led by T. Rey, the Lille pole will work with a Wallonian private for non-profit research center called Cenaero, which “provides to companies, involved in a technology innovation process, numerical simulation methods and tools to invent and design more competitive products. Internationally recognized, Cenaero is mainly active in the fields of aeronautical design, spacecrafts, manufacturing processes, and buildings and smart cities”. More details on the project can be found in Section 8.3.2.

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, 147-168
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• 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, 1093-1133
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• 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--1785
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• 4 articleC.Clément Cancès, T.Thomas Gallouët and L.Léonard Monsaingeon. Incompressible immiscible multiphase flows in porous media: a variational approach.Analysis & PDE1082017, 1845--1876
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• 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, 1525-1584
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• 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, 2473-2505
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• 7 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, 175-207
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• 8 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, 461-472
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• 9 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, 46-81
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• 10 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 Analysis2020
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• 11 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--680
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10.2 Publications of the year

10.3 Cited publications

• 50 techreportE.Etienne Ahusborde, B.Brahim Amaziane, A.Attila Baksay, G.Grażyna Bator, D.D. Becker, A.A. Bednar, M.M. Beres, R.R. Blaheta, Z.Z. Böhti, G.G. Bracke, L.L. Brazda, V.V. Brendler, K.Konstantin Brenner, J.J. Brezina, C.Clément Cancès, C.Claire Chainais-Hillairet, F.Florent Chave, F.Francis Claret, S.S. Domesova, V.Vaclava Havlova, M.M. Hokr, D.D. Horak, D.D. Jacques, F.F. Jankovsky, K.K. Kazymyrenko, O.O. Kolditz, T.T. Koudelka, T.T. Kovacs, T.T. Krejci, J.J. Kruis, E.E. Laloy, J.J. Landa, T.T. Lipping, D.D. Lukin, D.David Maš\'in, R.Roland Masson, J. C.J. C. L. Meeussen, M.M. Mollaali, A.A. Mon, L.L. Montenegro, V.Vanessa Montoya, G.Guillaume Pépin, J.J. Poonoosamy, N.N Prasianakis, Z.Z. Saâdi, J.J. Samper, G.Gianvito Scaringi, P.Pierre Sochala, C.Christophe Tournassat, K.K. Yoshioka and Y.Y. Yuankai. State Of the Art Report in the fields of numerical analysis and scientific computing. Final version as of 16/02/2020 deliverable D4.1 of the HORIZON 2020 project EURAD: European Joint Programme on Radioactive Waste Management.EURAD2020
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• 51 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, 1532-1567
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• 52 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, 133-153
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• 53 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 2018
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• 54 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 Media2005
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• 55 articleR.R. Araya, C.C. Harder, D.D. Paredes and F.F. Valentin. Multiscale Hybrid-Mixed method.SIAM J. Numer. Anal.5162013, 3505--3531
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• 56 articleA.A. Arnold, P.P. Markowich, G.G. Toscani and A.A. Unterreiter. On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations.Comm. Partial Differential Equations261-22001, 43-100URL: http://dx.doi.org/10.1081/PDE-100002246
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• 57 bookF.Franck Assous, P.Patrick Ciarlet Jr. and S.Simon Labrunie. Mathematical foundations of computational electromagnetism.198Applied Mathematical SciencesSpringer, Cham2018, ix+458URL: https://doi.org/10.1007/978-3-319-70842-3
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• 58 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, 333-340
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• 59 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 2020
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• 60 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--4467
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• 61 bookJ.J. Bear and Y.Y. Bachmat. Introduction to modeling of transport phenomena in porous media.4Springer1990
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• 62 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.
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• 63 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--214
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• 64 incollectionJ.-D.Jean-David Benamou, G.Guillaume Carlier and F.Filippo Santambrogio. Variational mean field games.Active Particles, Volume 1Springer2017, 141--171
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• 65 articleC.C. Besse. A relaxation scheme for the nonlinear Schrödinger equation.4232004, 934--952
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• 66 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, 147-168
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• 67 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/10.1137/130913432
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• 68 articleL.L. Boltzmann. Weitere Studien über das Wärmegleichgewicht unter Gasmolekl̎en.Wiener Berichte661872, 275--370
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• 69 articleY.Yann Brenier. Décomposition polaire et réarrangement monotone des champs de vecteurs.CR Acad. Sci. Paris Sér. I Math.3051987, 805--808
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• 70 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--139
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• 71 articleY.Yunhai Cai. Linear theory of microwave instability in electron storage rings.Physical Review Special Topics-Accelerators and Beams1462011, 061002
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• 72 articleJ.J. Cai and J.J. Shen. Two classes of linearly implicit local energy-preserving approach for general multi-symplectic Hamiltonian PDEs.4012020, 108975
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• 73 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.2020021
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• 74 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, 271-314
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• 75 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, 545-575
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• 76 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, 2544-2571
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• 77 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, 407-432
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• 78 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, 1123-1152
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• 79 articleC.Clément Cancès, T.Thomas Gallouët and G.Gabriele Todeschi. A variational finite volume scheme for Wasserstein gradient flows.Numerische Mathematik14632020, pp 437 - 480
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• 80 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. 2784-2710
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• 81 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--866
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• 82 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--1003
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• 83 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, 517-563
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• 84 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, 1-36
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• 85 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/s006050170032
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• 86 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/drl045
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• 87 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, 135-162
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• 88 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, 457-480
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• 89 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, 723-748
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• 90 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--1365
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• 91 bookO.O. Coussy. Poromechanics.John Wiley & Sons2004
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• 92 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, 2292–2300
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• 93 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--21
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• 94 articleA.A. Dragulescu and V. M.V. M. Yakovenko. Statistical mechanics of money.The European Physical Journal B-Condensed Matter and Complex Systems1742000, 723--729
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• 95 articleJ.J. Droniou. Finite volume schemes for diffusion equations: introduction to and review of modern methods.Math. Models Methods Appl. Sci.2482014, 1575-1620
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• 96 articleC.C Evain, C.C Szwaj, E.E Roussel, J.J Rodriguez, M.M Le Parquier, M.-A.M-A Tordeux, F.F Ribeiro, M.M Labat, N.N Hubert, J.-B.J-B Brubach and others. Stable coherent terahertz synchrotron radiation from controlled relativistic electron bunches.Nature Physics1572019, 635--639
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• 97 articleR.R. Eymard, T.T. Gallouët, C.C. Guichard, R.R. Herbin and R.R. Masson. TP or not TP, that is the question.Comput. Geosci.182014, 285--296
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• 98 articleF.Francis Filbet and T.Thomas Rey. A hierarchy of hybrid numerical methods for multi-scale kinetic equations.SIAM Journal on Scientific Computing373May 2015, A1218--A1247
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• 99 articleN.N. Fournier and S.S. Méléard. A Microscopic Probabilistic Description of a Locally Regulated Population and Macroscopic Approximations.Ann. Appl. Probab.1442004, 1880-1919
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• 100 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--60
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• 101 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, 99-149
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• 102 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--1106
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• 103 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/S003614450036757X
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• 104 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, 593-636
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• 105 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, 155-179
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• 106 articleR.R. Jordan, D.D. Kinderlehrer and F.F. Otto. The variational formulation of the Fokker-Planck equation.SIAM J. Math. Anal.2911998, 1--17
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• 107 articleL.L. Kantorovitch. On the translocation of masses.C. R. (Dokl.) Acad. Sci. URSS, n. Ser.371942, 199--201
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• 108 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--138
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• 109 articleP.P. Lafitte and G.G. Samaey. Asymptotic-preserving Projective Integration Schemes for Kinetic Equations in the Diffusion Limit.3422012, A579–-A602
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• 110 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--643
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• 111 articleS.Simon Lemaire. Bridging the hybrid high-order and virtual element methods.IMA J. Numer. Anal.4112021, 549--593
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• 112 articleH. C.H. C. Lie, K.K. Fackeldey and M.M. Weber. A square root approximation of transition rates for a Markov state model.SIAM J. Matrix Anal. Appl.342013, 738--756
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• 113 articleD.D. Matthes and S.S. Plazzotta. A variational formulation of the BDF2 method for metric gradient flows.ESAIM: M2AN5312019, 145-172
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• 114 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, 155-162
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• 115 articleA.A. Mielke. A gradient structure for reaction-diffusion systems and for energy-drift-diffusion systems.Nonlinearity2442011, 1329--1346URL: http://dx.doi.org/10.1088/0951-7715/24/4/016
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• 116 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.
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