2.1 Overall objectives

Together with the diffusion of scientific computing, there has been a recent and impressive increase of the demand for numerical methods. The problems to be addressed are everyday more complex and require specific numerical algorithms. The quality of the results has to be accurately assessed, so that in-silico experiments results can be trusted. Nowadays, producing such reliable numerical results goes way beyond the abilities of isolated researchers, and must be carried out by structured teams.

The topics addressed by the RAPSODI project-team belong to the broad theme of numerical methods for the approximation of solutions of systems of partial differential equations (PDEs). Besides standard convergence properties, a good numerical method for approximating a physical problem has to satisfy at least the following three criteria:

1. preservation at the discrete level of some crucial features of the solution, such as positivity of solutions, conservation of prescribed quantities (e.g. mass, the decay of physically motivated entropies etc.);
2. provide accurate numerical approximations at a reasonable computational cost (and ultimately maximize the accuracy at a fixed computational effort);
3. robustness with respect to physical conditions: the computational cost for a given accuracy should be essentially insensitive to change of physical parameters.

We aim to develop methods fulfilling the above quality criteria for physical models which all display a dissipative behavior, and that are motivated by industrial collaborations or multidisciplinary projects. In particular, we have identified a couple of specific situations we plan to investigate: models from corrosion science (in the framework of nuclear waste repository) 71, low-frequency electromagnetism 88, and mechanics of complex inhomogeneous fluids arising in avalanches 80 or in porous media 73.

Ideally, we should allow ourselves to design entirely new numerical methods. For some applications however (often in the context of industrial collaborations), the members of the team have to compose with existing codes. The numerical algorithms have thus to be optimized under this constraint.

2.2 Scientific context

Some technological bottlenecks related to points (a)–(c) mentioned above are well identified. In particular, it appears that a good numerical method should handle general meshes, so that dynamic mesh adaptation strategies can be used in order to achieve (b). But it should also be of the highest possible order while remaining stable in the sense of (a), and robust in the sense of (c). There have been numerous research contributions on each point of (a)–(c) in the last decades, in particular for solving each difficulty separately, but combining them still leads to unsolved problems of crucial interest.

Let us mention for example the review paper by J. Droniou 97, where it is highlighted that all the linear methods for solving diffusion equations on general meshes suffer from the same lack of monotonicity and preserve neither the positivity of the solutions nor the decay of the entropy. Moreover, there is no complete convergence proof for the nonlinear methods exposed in 97. The first convergence proof for a positivity preserving and entropy diminishing method designed to approximate transient dissipative equations on general meshes was proposed very recently in 81. The idea and the techniques introduced in 81 should be extended to practical applications.

In systems of PDEs, the values of the physical parameters often change the qualitative behavior of the solutions. Then, one challenge in the numerical approximation of such systems is the design of methods which can be applied for a large range of parameters, as in particular in the regime of singular perturbations. Such schemes, called asymptotic-preserving (AP) schemes 103, are powerful tools as they allow the use of the same scheme for a given problem and for its limit with fixed discretization parameters. In many cases, the AP property of numerical schemes is just empirically established, without any rigorous proof. We aim to extend the techniques recently introduced in 77 for the drift-diffusion system, and based on the control of the numerical dissipation of entropy, to other dissipative systems in order to prove the AP property of numerical schemes.

The question of the robustness of the numerical methods with respect to the physical parameters is also fundamental for fluid mixture models. The team already developed such schemes for the variable density Navier–Stokes system 79, 80. We aim to propose new ones for more complex models with the same philosophy in mind. On the one hand, we will be interested in high-order schemes, which must be as simple as possible in view of 3D practical implementations. Let us stress that combining high order accuracy and stability is very challenging. On the other hand, the optimization of the computation will have to be considered, in particular with the development of some a posteriori error estimators. Impressive progresses have been achieved in this field 94, allowing important computational savings without compromising the accuracy of the results. Recently, we successfully applied this strategy to the Reissner–Mindlin model arising in solid mechanics 90, the dead-oil model for porous media flows 83, or the Maxwell equations in their quasi-static approximation for some eddy current problems 88, 89. We aim to develop new a posteriori estimators for other dissipative systems, like fluid mixture models.

In a nutshell, our goal is to take advantage of and extend the most recent breakthroughs of the mathematical community to tackle in an efficient way some application-guided problems coming either from academics or from industrial partners. To this end, we shall focus on the following objectives, which are necessary for the applications we have in mind:

1. Design and analysis of structure-preserving numerical methods.
2. Computational optimization.

3.1 Design and analysis of structure-preserving schemes

3.2 Optimizing the computational efficiency

4.1 Porous media flows

Porous media flows are of great interest in many contexts, like, e.g. oil engineering, water resources management, nuclear waste repository management, or carbon dioxide sequestration. We refer to 73, 72 for an extensive discussion on porous media flow models.

From a mathematical point of view, the transport of complex fluids in porous media often leads to possibly degenerate parabolic conservation laws. The porous rocks can be highly heterogeneous and anisotropic. Moreover, the grids on which one intends to solve numerically the problems are prescribed by the geological data, and might be nonconformal with cells of various shapes. Therefore, the schemes used for simulating such complex flows must be particularly robust.

4.2 Corrosion and concrete carbonation

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.

From a mathematical point of view, the modeling of concrete carbonation (see 67) as the modeling of corrosion in an underground repository (DPCM model developed by Bataillon et al. 71) lead to systems of PDEs posed on moving domains. The coupling between convection-diffusion-reaction equations and moving boundary equations leads to challenging mathematical questions.

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 Stratigraphy

The knowledge of the geology is a prerequisite before simulating flows within the subsoil. Numerical simulations of the geological history thanks to stratigraphy numerical codes allow to complete the knowledge of the geology where experimental data are lacking. Stratigraphic models consist in a description of the erosion and sedimentation phenomena at geological scales.

The characteristic time scales for the sediments are much larger than the characteristic time scales for the water in the river. However, the (time-averaged) water flux plays a crucial role in the evolution of the stratigraphy. Therefore, defining appropriate models that take the coupling between the rivers and the sediments into account is fundamental and challenging. Once the models are at hand, efficient numerical methods must be developed.

4.5 Low-frequency electromagnetism

Numerical simulation is nowadays an essential tool in order to design electromagnetic systems, by estimating the electromagnetic fields generated in a wide variety of devices. An important challenge for many applications is to quantify the intensity of the electric field induced in a conductor by a current generated in its neighborhood. In the low-frequency regime, we can for example cite the study of the impact on the human body of a high-tension line or, for higher frequencies, the one of a smartphone. But the ability to simulate accurately some electromagnetic fields is also very useful for nondestructive control, in the context of the maintenance of nuclear power stations for example. The development of efficient numerical tools, among which a posteriori error estimators, is consequently necessary to reach a high precision of calculation in order to provide as reliable estimations as possible.

5.1 Platform ParaSkel++

ParaSkel++ is a parallel C++ platform for the optimized, arbitrary-order, 2/3D numerical approximation of PDEs on general meshes using skeletal Galerkin methods.

Skeletal Galerkin methods form a vast class of numerical methods for the approximation of PDEs that is based on the following 2 building principles:

• $•$ the degrees of freedom (DOF) of the method split into (i) skeletal DOF, attached to the geometrical entities composing the mesh skeleton (vertices, edges, faces) and common, locally, to all cells sharing the geometrical entity in question, which prescribe the conformity properties of the underlying functional space, and (ii) bulk DOF, attached to the interior of the cell, that are not shared between adjacent elements and which play no role in the prescription of the conformity properties of the functional space;
• $•$ the global discrete bilinear form writes as the sum of strictly local (to each cell) contributions.

The very structure of skeletal methods grants them the property of being amenable to static condensation, i.e. bulk DOF can be locally eliminated in terms of the skeletal DOF of the cell in question. The final global system to solve thus writes in terms of the skeletal DOF only, whence the vocable of “skeletal” method. Due to the local character of the computations to perform to assemble the system, skeletal methods are highly parallelizable. The class of skeletal methods encompasses in particular (conforming and nonconforming) FE methods, and virtual-like Galerkin methods (VEM, HDG, HHO...). It does not contain DG methods.

The ParaSkel++ platform offers an optimized and unified C++ architecture for the implementation of arbitrary-order skeletal Galerkin methods on general 2/3D meshes. The development of this platform has begun in February 2020, and is expected to last (at least) 30 months. The P.I. of the project is S. Lemaire, and the main developer is L. Beaude.

5.2 Other codes

The code FPmuRNA, that has been used to produce the numerical simulations of the article “A Fokker–Planck approach to the study of robustness in gene expression” 30, has been published online as open-source (available at gitlab.inria.fr/herda/fpmurna). FPmuRNA is a Matlab code developed by M. Herda.

The code KinDiff, that has been used to produce the numerical simulations of the article “Hypocoercivity and diffusion limit of a Finite Volume scheme for linear kinetic equations” 15, has been published online as open-source (available at gitlab.com/thoma.rey/FV_HipoDiff). KinDiff, developed by M. Bessemoulin-Chatard (CNRS and Université de Nantes), M. Herda, and T. Rey, consists in a Python code and a Jupyter Notebook. It approximates the solutions to the kinetic Fokker–Planck and linear BGK equations in a unidimensional periodic domain in space and a symmetric bounded unidimensional domain in velocity. Its outputs are the discretization of the distribution function in the phase space as well as the moments, along with some visualizations of these quantities. Kinetic equations display asymptotic behaviors such as return to equilibrium in large time, or convergence towards macroscopic equations in the diffusive limit. The Finite Volume method implemented within KinDiff reproduces these asymptotics at the discrete level.

The code 99, that has been used to produce the numerical simulations of the article “Finite Volumes for the Stefan–Maxwell cross-diffusion system” 50, has been published online as open-source (available at 10.5281/zenodo.3934285). This Julia code, developed by C. Cancès, V. Ehrlacher, and L. Monasse, simulates the Stefan–Maxwell equations on Cartesian grids thanks to an entropy-diminishing Two-Point Flux Approximation Finite Volume scheme.

The code that has been used to perform the computer-assisted proofs of the article “Existence of traveling wave solutions for the Diffusion Poisson Coupled Model: a computer-assisted proof” 48 has been published online as open-source (available at sites.google.com/site/maximebreden/research). This Matlab code has been developed by M. Breden, C. Chainais-Hillairet, and A. Zurek.

6.1 Modeling and numerical simulation of complex fluids

In 45, C. Cancès et al. establish an error estimate, within the generic framework for the spatial discretisation of partial differential equations of the Gradient Discretisation Method (GDM), for a class of degenerate parabolic problems. This result is obtained under very mild regularity assumptions on the exact solution. Their study covers well-known models like the porous medium equation and the fast diffusion equations, as well as the strongly degenerate Stefan problem. Several schemes are then compared in a last section devoted to numerical results.

In 40, C. Cancès et al. propose a $P^1$ Finite Element scheme with mass-lumping for a model of two incompressible and immiscible phases in a porous media flow. They prove the dissipation of the free energy and the existence of a solution to the nonlinear scheme. They also present numerical simulations to illustrate the behavior of the scheme.

In 51, 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 deGennes. 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 82 is the nonlinearity of the energy density in the gradient part, which becomes singular at the interface between pure and mixed phases.

In 23, C. Cancès, N. Peton et al. propose a new water flow-driven forward stratigraphic model. Stratigraphy is a discipline of physics that aims at predicting the geological composition of the subsoil. The model enjoys the following particularities. First, the water surface flow is modelled at the continuous level, in opposition to what is currently done in this community. Second, the model incorporates a constraint on the erosion rate. A stable numerical scheme is proposed to simulate the model.

In 17, C. Calgaro, C. Colin, and E. Creusé propose a combined Finite Volume-Finite Element scheme for the solution of a specific low-Mach model expressed in the velocity, pressure and temperature variables. The dynamic viscosity of the fluid is given by an explicit function of the temperature, leading to the presence of a so-called Joule term in the mass conservation equation. First, they prove a discrete maximum principle for the temperature. Second, the numerical fluxes defined for the Finite Volume computation of the temperature are efficiently derived from the discrete Finite Element velocity field obtained by the solution of the momentum equation. Several numerical tests are presented to illustrate the theoretical results and to underline the efficiency of the scheme in terms of convergence rates.

In 37, C. Calgaro and E. Creusé introduce a Finite Volume method to approximate the solution of a convection-diffusion equation involving a Joule term. They propose a way to discretize this so-called “Joule effect” term in a consistent manner with respect to the nonlinear diffusion one, in order to ensure some maximum principle properties on the solution. They investigate the numerical behavior of the scheme on two original benchmarks.

In 53, T. Rey et al. review recent mathematical results in kinetic granular materials, especially for those which arose since the last review by Villani on the same subject. This model describes the nonequilibrium behavior of materials composed of a large number of interacting, nonnecessarily microscopic particles, such as grains or planetary rings. This theoretical knowledge is then used to validate a new high-order numerical method for this equation, highlighting through numerics some theoretical open problems.

In 30, M. Herda et al. study several Fokker–Planck equations arising from a stochastic chemical kinetic system modeling a gene regulatory network in biology. The densities solving the Fokker–Planck equations describe the joint distribution of the messenger RNA and micro RNA content in a cell. They provide theoretical and numerical evidences that the robustness of the gene expression is increased in the presence of micro RNA. At the mathematical level, increased robustness shows in a smaller coefficient of variation of the marginal density of the messenger RNA in the presence of micro RNA. These results follow from explicit formulas for solutions. Moreover, thanks to dimensional analyses and numerical simulations they provide qualitative insight into the role of each parameter in the model. As the increase of gene expression level comes from the underlying stochasticity in the models, they eventually discuss the choice of noise in their models and its influence on their results.

In 48, C. Chainais-Hillairet, A. Zurek et al. present and apply a computer-assisted method in order to prove the existence of traveling wave solutions to the Diffusion Poisson Coupled Model arising in corrosion modeling. They also establish a precise and certified description of the solutions.

6.2 Numerical simulation in low-frequency electromagnetism

In 29, 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, existence and uniqueness of the weak solution to the $𝐀-\phi -𝐁$ system is deduced.

In 43, F. Chave, S. Lemaire et al. introduce a three-dimensional Hybrid High-Order (HHO) method for the magnetostatics problem. The proposed method is easy to implement, supports general polyhedral meshes, and allows for arbitrary orders of approximation.

In 54, F. Chave, 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 study two HHO 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 with star-shaped elements, 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 Structure-preserving numerical methods

In 41, C. Chainais-Hillairet and M. Herda apply an iterative energy method à la de Giorgi in order to establish $L^{\infty }$ bounds for numerical solutions of noncoercive convection-diffusion equations with mixed Dirichlet-Neumann boundary conditions.

In 36, C. Chainais-Hillairet et al. propose some Finite Volume schemes for unipolar energy-transport models. Using a reformulation in dual entropy variables, they show the decay of a discrete entropy with control of the discrete entropy dissipation.

In 47, C. Chainais-Hillairet et al. develop their work 36. They establish a priori estimates which lead to the existence of a solution to the scheme, and they prove the exponential decay of the discrete relative entropy towards the thermal equilibrium. Moreover, numerical results assess the good behavior of the numerical schemes.

In 19, C. Cancès, C. Chainais-Hillairet, B. Gaudeul et al. consider an unipolar degenerate drift-diffusion system arising in the modeling of organic semiconductors. They design four different Finite Volume schemes based on four different formulations of the fluxes. They provide a stability analysis and existence results for the four schemes; the convergence is established for two of them.

In 38, C. Cancès, C. Chainais-Hillairet, B. Gaudeul et al. consider an unipolar degenerate drift-diffusion system where the relation between the concentration of the charged species $c$ and the chemical potential $h$ is $h\left(c\right)=log\frac{c}{1-c}$. For four different Finite Volume schemes based on four different formulations of the fluxes of the problem, they discuss stability and existence results. For two of them, they report a convergence proof. Numerical experiments illustrate the behavior of the different schemes.

In 26, C. Cancès et al. propose a Finite Element scheme for the numerical approximation of degenerate parabolic problems in the form of a nonlinear anisotropic Fokker–Planck equation. The scheme is energy-stable, only involves physically motivated quantities in its definition, and is able to handle general unstructured grids. Its convergence is rigorously proven thanks to compactness arguments, under very general assumptions. Although the scheme is based on Lagrange Finite Elements of degree 1, it is locally conservative after a local post-processing giving rise to an equilibrated flux. This also allows to derive a guaranteed a posteriori error estimate for the approximate solution. Numerical experiments are presented in order to give evidence of a very good behavior of the proposed scheme in various situations involving strong anisotropy and drift terms.

In 25, C. Cancès et al. study a time-implicit Finite Volume scheme for the degenerate Cahn–Hilliard model proposed in 98 and studied mathematically in 82. The scheme is shown to preserve the key properties of the continuous model, namely mass conservation, positivity of the concentrations, the decay of the energy, and the control of the entropy dissipation rate. This allows to establish the existence of a solution to the nonlinear algebraic system corresponding to the scheme. Furthermore, C. Cancès et al. show thanks to compactness arguments that the approximate solution converges towards a weak solution to the continuous problem as the discretization parameters tend to 0. Numerical results illustrate the behavior of the numerical scheme.

In 39, C. Cancès and B. Gaudeul propose a Two-Point Flux Approximation Finite Volume scheme for the approximation of the solutions to an entropy dissipative cross-diffusion system. The scheme is shown to preserve several key properties of the continuous system, among which positivity and decay of the entropy. Numerical experiments illustrate the behavior of the scheme.

In 22, C. Cancès and B. Gaudeul study a Two-Point Flux Approximation Finite Volume scheme for a cross-diffusion system. The scheme is shown to preserve the key properties of the continuous system, among which the decay of the entropy. The convergence of the scheme is established thanks to compactness properties based on the discrete entropy-entropy dissipation estimate. Numerical results illustrate the behavior of the scheme.

In 50, 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 21, C. Cancès et al. propose a variational Finite Volume scheme for the computation of Wasserstein gradient flows. The discrete solution is the minimizer of a discrete action, keeping track at the discrete level of the optimal character of the gradient flow. The spatial discretization relies on upstream mobility fluxes, while an implicit linearization of the Wasserstein distance is used in order to reduce the computational cost by avoiding an inner time-stepping as in the related contributions of the literature.

In 13, R. Bailo et al. study an implicit Finite Volume scheme for nonlinear, nonlocal aggregation-diffusion equations which exhibit a gradient flow structure, recently introduced in 70. Crucially, this scheme keeps the dissipation property of an associated fully discrete energy, and does so unconditionally with respect to the time step. The main contribution in this work is to show the convergence of the method under suitable assumptions on the diffusion functions and potentials involved.

In 57, A. Natale et al. construct and analyze Two-Point Flux Approximation Finite Volume discretizations of the quadratic optimal transport problem in its dynamic form. They show numerically that these types of discretizations are prone to form instabilities in their more natural implementation, and propose a variation based on nested meshes in order to overcome these issues. Moreover, they introduce a strategy based on the barrier method to solve the discrete optimization problem.

In 58, T. Rey et al. focus on the stability properties of some recently introduced spectral methods that preserve equilibrium. Thanks to the high accuracy and the possibility to use fast algorithms, such methods represent an effective way to approximate the Boltzmann collision operator. On the other hand, the loss of some local invariants usually leads to the wrong long-time behavior. In this paper, using the perturbation argument developed by Filbet and Mouhot for the homogeneous Boltzmann equation, the authors prove stability, convergence, and spectrally accurate long-time behavior of the equilibrium-preserving approach.

In 44, T. Rey et al. present a new Finite Volume method for computing numerical approximations of a system of nonlocal transport equations modeling interacting species. In this work, the nonlocal continuity equations are treated as conservative transport equations with a nonlocal, nonlinear, rough velocity field. Some properties of the method are analyzed, and numerical simulations are performed.

In 14, I. Lacroix-Violet et al. focus on the numerical integration in time of nonlinear Schrödinger equations using different methods preserving the energy or a discrete analog of it. In particular, they give a rigorous proof of the order of the relaxation method (presented in 76 for cubic nonlinearities) and they propose a generalized version that allows to deal with general power law nonlinearities. Numerical simulations for different physical models show the efficiency of these methods.

6.4 Cost reduction for numerical methods

In 34, S. Lemaire presents a unifying viewpoint on Hybrid High-Order 93 and Virtual Element 74 methods on general polytopal meshes in dimension 2 or 3, in terms of both formulation and analysis. The focus is on a model Poisson problem. To bridge the two paradigms, (i) he transcribes the (conforming) Virtual Element method into the Hybrid High-Order framework and (ii) proves $H^m$ approximation properties for the local polynomial projector in terms of which the local Virtual Element discrete bilinear form is defined. This allows him to perform a unified analysis of Virtual Element/Hybrid High-Order methods, that differs from standard Virtual Element analyses by the fact that the approximation properties of the underlying virtual space are not explicitly used. As a complement to this unified analysis, he also studies interpolation in local virtual spaces, shedding light on the differences between the conforming and nonconforming cases.

In 55, I. Lacroix-Violet et al. introduce a new class of numerical methods for the time integration of evolution equations set as Cauchy problems of ODEs or PDEs. The systematic design of these methods mixes the Runge–Kutta collocation formalism with collocation techniques, in such a way that the methods are linearly implicit and have high order. The fact that these methods are implicit allows to avoid CFL conditions when the large systems to integrate come from the space discretization of evolution PDEs. Moreover, these methods are expected to be efficient since they only require to solve one linear system of equations at each time step, and efficient techniques from the literature can be used to do so.

In 35, S. Bassetto, C. Cancès et al. propose an efficient nonlinear solver for the resolution of the Richards equation. It is based on variable switching and can be easily implemented thanks to a fictitious variable allowing to describe both the saturation and the pressure. Numerical experiments show that the method enables to use Newton’s method with large time steps, a reasonable number of iterations, and in regions where the pressure-saturation relationship is given by a graph.

In 24, C. Cancès and D. Maltese propose a reduced model for the migration of hydrocarbons in heterogeneous porous media. Their model keeps track of the time variable. This allows to compute steady-states that cannot be reached by the commonly used ray-tracing and invasion-percolation algorithms. An efficient Finite Volume scheme allowing for very large time steps is then proposed.

6.5 Asymptotic analysis

In 27, C. Chainais-Hillairet and M. Herda study the large-time behavior of the solutions to Finite Volume discretizations of convection-diffusion equations or systems endowed with non-homogeneous Dirichlet and Neumann type boundary conditions. Their results concern various linear and nonlinear models such as Fokker–Planck equations, porous media equations, or drift-diffusion systems for semiconductors. For all of these models, some relative entropy principle is satisfied and implies exponential decay to the stationary state. They show that in the framework of Finite Volume schemes on orthogonal meshes, a large class of two-point monotone fluxes preserves this exponential decay of the discrete solution to the discrete steady-state of the scheme.

In 20, C. Cancès, C. Chainais-Hillairet, M. Herda et al. analyze the large-time behavior of a family of nonlinear Finite Volume schemes for anisotropic convection-diffusion equations set in a bounded bidimensional domain and endowed with either Dirichlet and/or no-flux boundary conditions. They show that the solutions to the Two-Point Flux Approximation (TPFA) and Discrete Duality Finite Volume (DDFV) schemes under consideration converge exponentially fast towards their steady-state. The analysis relies on discrete entropy estimates and discrete functional inequalities. As a by-product of their analysis, they establish new discrete Poincaré–Wirtinger, Beckner and logarithmic Sobolev inequalities. Their theoretical results are illustrated by numerical simulations.

In 42, C. Chainais-Hillairet et al. introduce a nonlinear DDFV scheme for an anisotropic linear convection-diffusion equation with mixed boundary conditions and establish the exponential decay of the scheme towards its steady-state.

In 15, M. Herda, T. Rey et al. are interested in the asymptotic analysis of a Finite Volume scheme for one-dimensional linear kinetic equations, with either Fokker–Planck or linearized BGK collision operator. Thanks to appropriate uniform estimates, they establish that the proposed scheme is asymptotic-preserving in the diffusive limit. Moreover, they adapt to the discrete framework the hypocoercivity method proposed in 95 to prove the exponential return to equilibrium of the approximate solution. They obtain decay estimates that are uniform in the diffusive limit. Finally, they present an efficient implementation of the proposed numerical schemes, and perform numerous numerical simulations assessing their accuracy and efficiency in capturing the correct asymptotic behaviors of the models.

In 12, M. Herda et al. are interested in the large-time behavior of linear kinetic equations with heavy-tailed local equilibria. Their main contribution concerns the kinetic Lévy–Fokker–Planck equation, for which they adapt hypocoercivity techniques in order to show that solutions converge exponentially fast to the global equilibrium. Compared to the classical kinetic Fokker–Planck equation, the issues here concern the lack of symmetry of the nonlocal Lévy–Fokker–Planck operator and the understanding of its regularization properties. As a complementary related result, they also treat the case of the heavy-tailed BGK equation.

In 49, I. Lacroix-Violet et al. consider global weak solutions to compressible Navier–Stokes–Korteweg equations with density dependent viscosities, in a periodic domain $\Omega ={𝕋}^3$, with a linear drag term with respect to the velocity. The main result concerns the exponential decay to equilibrium of such solutions using log-Sobolev type inequalities. In order to show such a result, the starting point is a global weak-entropy solutions definition, introduced in 78. Assuming extra assumptions on the shear viscosity when the density is close to vacuum and when the density tends to infinity, I. Lacroix-Violet et al. conclude the exponential decay to equilibrium. The result also covers the quantum Navier–Stokes system with a drag term.

In 16, T. Rey et al. propose a new mathematical model intended to describe dynamically the evolution of knowledge in structured societies of interacting individuals. This process, termed cumulative culture, has been extensively studied by evolutionary anthropologists, both theoretically and experimentally. Some of the mathematical properties of the new model are analyzed, and exponential convergence towards a global equilibrium is shown for a simplified model. A numerical method is finally proposed to simulate the complete model.

In 11, following the ideas of V. V. Zhikov and A. L. Pyatnitskii, and more precisely the stochastic two-scale convergence, B. Merlet et al. establish a homogenization theorem in a stochastic setting for two nonlinear equations: the equation of harmonic maps into the sphere, and the Landau–Lifshitz equation. Homogenization results for nonlinear problems are known to be difficult. In this particular case, the equations have strong nonlinear features; in particular, in general, their solutions are not unique. Here, the authors take advantage of the different equivalent definitions of weak solutions to the nonlinear problem to apply typical linear homogenization recipes.

6.6 Applied calculus of variations

In 32, B. Merlet et al. study a variational problem which models the behavior of topological singularities on the surface of a biological membrane in $P_{\beta }$-phase (see 107). The problem combines features of the Ginzburg–Landau model in 2D and of the Mumford–Shah functional. As in the classical Ginzburg–Landau theory, a prescribed number of point vortices appears in the moderate energy regime; the model allows for discontinuities, and the energy penalizes their length. The novel phenomenon here is that the vortices have a fractional degree $1/m$ with $m$ prescribed. Those vortices must be connected by line discontinuities to form clusters of total integer degrees. The vortices and line discontinuities are therefore coupled through a topological constraint. As in the Ginzburg–Landau model, the energy is parameterized by a small length scale $\epsilon >0$. B. Merlet et al. perform a complete $\Gamma$-convergence analysis of the model as $\epsilon \downarrow 0$ in the moderate energy regime. Then, they study the structure of minimizers of the limit problem. In particular, the line discontinuities of a minimizer solve a variant of the Steiner problem.

In 28, B. Merlet et al. establish new results on the approximation of $k$-dimensional surfaces ($k$-rectifiable currents) by polyhedral surfaces with convergence in $h$-mass and with preservation of the boundary (the approximating polyhedral surface has the same boundary as the limit). This approximation result is required in the convergence study of 87.

In 33, B. Merlet et al. study a family of functionals penalizing oblique oscillations. These functionals naturally appear in some variational problems related to pattern formation and are somewhat reminiscent of those introduced by Bourgain, Brezis and Mironescu to characterize Sobolev functions. More precisely, for a function $u$ defined on a tensor product ${\Omega }_1\times {\Omega }_2$, the family of functionals ${\left\{E_{\epsilon }\left(u\right)\right\}}_{\epsilon >0}$ that they consider vanishes if $u$ is of the form $u\left(x_1\right)$ or $u\left(x_2\right)$. They prove the converse property and related quantitative results. In particular, they describe the fine properties of functions with ${sup}_{\epsilon }{E}_{\epsilon }\left(u\right)<\infty$ by showing that, roughly, such $u$ is piecewise of the form $u\left(x_1\right)$ or $u\left(x_2\right)$ on domains separated by lines where the energy concentrates. It turns out that this problem naturally leads to the study of various differential inclusions, and has connections with branched transportation models.

In 59, M. Pegon studies large volume minimizers of isoperimetric problems derived from Gamow's liquid drop model for the atomic nucleus, involving the competition of a perimeter term and repulsive nonlocal potentials. Considering a large class of potentials, given by general radial nonnegative kernels which are integrable on ${ℝ}^n$, such as Bessel potentials, M. Pegon proves the existence of minimizers of arbitrarily large mass, provided that the first moment of the kernels is below an explicit threshold. This contrasts with the case of Riesz potentials, where minimizers do not exist above a critical mass. In addition, renormalizing to a fixed volume, any sequence of minimizers converges to the ball as the mass goes to infinity. Finally, M. Pegon shows that the threshold on the first moment of the kernels is sharp, in the sense that large balls go from stable to unstable. A direct consequence of the instability of large balls above this threshold is that there exist nontrivial compactly supported kernels for which the problems admit minimizers which are not balls, that is, symmetry breaking occurs.

6.7 Approximation theory

In 31, M. Herda et al. propose an iterative algorithm for the numerical computation of sums of squares of polynomials approximating given data at prescribed interpolation points. The method is based on the definition of a convex functional $G$ arising from the dualization of a quadratic regression over the Cholesky factors of the sum of squares decomposition. In order to justify the construction, the domain of $G$, the boundary of the domain, and the behavior at infinity are analyzed in details. When the data interpolate a positive univariate polynomial, M. Herda et al. show that in the context of the Lukacs sum of squares representation, $G$ is coercive and strictly convex, which yields a unique critical point and a corresponding decomposition in sum of squares. For multivariate polynomials which admit a decomposition in sum of squares, and up to a small perturbation of size $\epsilon$, $G^{\epsilon }$ is always coercive, and so its minimum yields an approximate decomposition in sum of squares. Various unconstrained descent algorithms are proposed to minimize $G$. Numerical examples are provided, for univariate and bivariate polynomials.

In 18, M. Herda et al. investigate the numerical approximation of bounded functions by polynomials satisfying the same bounds. The contribution makes use of the recent algebraic characterization found in 92, 91, where an interpretation of monovariate polynomials with two bounds is provided in terms of a quaternion algebra and the Euler four-squares formulas. Thanks to this structure, M. Herda et al. generate a new nonlinear projection algorithm onto the set of polynomials with two bounds. The numerical analysis of the method provides theoretical error estimates showing stability and continuity of the projection. Some numerical tests illustrate this novel algorithm for constrained polynomial approximation.

7.1 Bilateral contracts with industry

The PhD thesis of S. Bassetto is funded by IFPEn. The contract follows the lines of the bilateral contract between Inria and IFPEn.

7.2 Bilateral grants with industry

CEA (C. Bataillon) and ANDRA (L. 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 can be found in Section 8.2.1.

8.1 International research visitors

8.2 European initiatives

8.3 National initiatives

8.4 Regional initiatives

9.1 Promoting scientific activities

9.2 Teaching - Supervision - Juries

9.3 Popularization

10.1 Major publications

• 1 articleMarianneM. Bessemoulin-Chatard and ClaireC. Chainais-Hillairet. Exponential decay of a finite volume scheme to the thermal equilibrium for drift--diffusion systemsJournal of Numerical Mathematics2532017, 147-168
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• 2 articleCaterinaC. Calgaro, EmmanuelE. Creusé, ThierryT. Goudon and StellaS. Krell. Simulations of non homogeneous viscous flows with incompressibility constraintsMathematics and Computers in Simulation1372017, 201-225
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• 3 articleClémentC. Cancès, ThomasT. Gallouët and LéonardL. Monsaingeon. Incompressible immiscible multiphase flows in porous media: a variational approachAnalysis & PDE1082017, 1845--1876
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• 4 articleClémentC. Cancès and CindyC. Guichard. Numerical analysis of a robust free energy diminishing Finite Volume scheme for parabolic equations with gradient structureFoundations of Computational Mathematics1762017, 1525-1584
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• 5 inproceedingsClaireC. Chainais-Hillairet, BenoîtB. Merlet and AlexisA. Vasseur. Positive Lower Bound for the Numerical Solution of a Convection-Diffusion EquationFVCA8 2017 - International Conference on Finite Volumes for Complex Applications VIIILille, FranceSpringerJune 2017, 331-339
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• 6 articleDaniele AntonioD. Di Pietro, AlexandreA. Ern and SimonS. Lemaire. An arbitrary-order and compact-stencil discretization of diffusion on general meshes based on local reconstruction operatorsComputational Methods in Applied Mathematics144June 2014, 461-472
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• 7 articleGiacomoG. Dimarco, RaphaëlR. Loubère, JacekJ. Narski and ThomasT. Rey. An efficient numerical method for solving the Boltzmann equation in multidimensionsJournal of Computational Physics3532018, 46-81
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• 8 articleFrancisF. Filbet and MaximeM. Herda. A finite volume scheme for boundary-driven convection-diffusion equations with relative entropy structureNumerische Mathematik13732017, 535-577
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• 9 articleIngridI. Lacroix-Violet and AlexisA. Vasseur. Global weak solutions to the compressible quantum Navier--Stokes equation and its semi-classical limitJournal de Mathématiques Pures et Appliquées1142018, 191-210
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• 10 articleBenôitB. Merlet. A highly anisotropic nonlinear elasticity model for vesicles I. Eulerian formulation, rigidity estimates and vanishing energy limitArch. Ration. Mech. Anal.21722015, 651--680
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10.2 Publications of the year

10.3 Other

10.4 Cited publications

• 63 articleR. Abgrall. A review of residual distribution schemes for hyperbolic and parabolic problems: the July 2010 state of the artCommun. Comput. Phys.1142012, 1043--1080
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• 64 articleR. Abgrall, G. Baurin, A. Krust, D. de Santis and M. Ricchiuto. Numerical approximation of parabolic problems by residual distribution schemesInternat. J. Numer. Methods Fluids7192013, 1191--1206
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• 65 articleR. Abgrall, A. Larat and M. Ricchiuto. Construction of very high order residual distribution schemes for steady inviscid flow problems on hybrid unstructured meshesJ. Comput. Phys.230112011, 4103--4136
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• 66 articleR. Abgrall, A. Larat, M. Ricchiuto and C. Tavé. A simple construction of very high order non-oscillatory compact schemes on unstructured meshesComput. & Fluids3872009, 1314--1323
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• 67 articleToyohikoT. Aiki and AdrianA. Muntean. A free-boundary problem for concrete carbonation: front nucleation and rigorous justification of the $\sqrt{t}$-law of propagation'Interfaces Free Bound.1522013, 167--180
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• 68 articleB. Amaziane, A. Bergam, M. El Ossmani and Z. Mghazli. A posteriori estimators for vertex centred finite volume discretization of a convection-diffusion-reaction equation arising in flow in porous mediaInternat. J. Numer. Methods Fluids5932009, 259--284URL: http://dx.doi.org/10.1002/fld.1456
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• 69 articleI. Babuška and W. C.W. Rheinboldt. Error estimates for adaptive finite element computationsSIAM J. Numer. Anal.1541978, 736--754
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• 70 articleRafaelR. Bailo, José A.J. Carrillo and JingweiJ. Hu. Fully Discrete Positivity-Preserving and Energy-Dissipating Schemes for Aggregation-Diffusion Equations with a Gradient Flow StructureCommun. Math. Sci.1852020, 1259--1303
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• 71 articleC. Bataillon, F. Bouchon, C. Chainais-Hillairet, C. Desgranges, E. Hoarau, F. Martin, S. Perrin, M. Tupin and J. Talandier. Corrosion modelling of iron based alloy in nuclear waste repositoryElectrochim. Acta55152010, 4451--4467
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• 72 book J. Bear and Y. Bachmat. Introduction to modeling of transport phenomena in porous media 4 Springer 1990
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• 73 book J. Bear. Dynamic of Fluids in Porous Media New York American Elsevier 1972
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• 74 articleL. Beirão da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L.~D.L. Marini and A. Russo. Basic principles of virtual element methodsMath. Models Methods Appl. Sci. (M3AS)2312013, 199--214
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• 75 articleS. Berrone, V. Garbero and M. Marro. Numerical simulation of low-Reynolds number flows past rectangular cylinders based on adaptive finite element and finite volume methodsComput. & Fluids402011, 92--112URL: http://dx.doi.org/10.1016/j.compfluid.2010.08.014
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• 76 phdthesis ChristopheC. Besse. Analyse numérique des systèmes de Davey--Stewartson Université Bordeaux 1 1998
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• 77 article MarianneM. Bessemoulin-Chatard, ClaireC. Chainais-Hillairet and Marie-HélèneM.-H. Vignal. Study of a fully implicit scheme for the drift-diffusion system. Asymptotic behavior in the quasi-neutral limit SIAM, J. Numer. Anal. 52 4 2014
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• 78 unpublishedD. Bresch, A. Vasseur and C. Yu. Global Existence of Entropy-Weak Solutions to the Compressible Navier-Stokes Equations with Non-Linear Density Dependent Viscosities2019, working paper or preprintURL: https://arxiv.org/abs/1905.02701
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• 79 articleC. Calgaro, E. Creusé and T. Goudon. An hybrid finite volume-finite element method for variable density incompressible flowsJ. Comput. Phys.22792008, 4671--4696
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• 80 articleCaterinaC. Calgaro, EmmanuelE. Creusé and ThierryT. Goudon. Modeling and simulation of mixture flows: application to powder-snow avalanchesComput. & Fluids1072015, 100--122URL: http://dx.doi.org/10.1016/j.compfluid.2014.10.008
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• 81 articleClémentC. Cancès and CindyC. Guichard. Convergence of a nonlinear entropy diminishing Control Volume Finite Element scheme for solving anisotropic degenerate parabolic equationsMathematics of Computation852982016, 549-580
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• 82 articleC. Cancès, D. Matthes and F. Nabet. A two-phase two-fluxes degenerate Cahn--Hilliard model as constrained Wasserstein gradient flowArch. Ration. Mech. Anal.23322019, 837--866
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• 83 articleC. Cancès, I. S.I. Pop and M. Vohral\'\ik. An a posteriori error estimate for vertex-centered finite volume discretizations of immiscible incompressible two-phase flowMath. Comp.832852014, 153--188URL: http://dx.doi.org/10.1090/S0025-5718-2013-02723-8
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• 84 articleJ. A.J. Carrillo, A. Jüngel, P. A.P. Markowich, G. Toscani and A. Unterreiter. Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalitiesMonatsh. Math.13312001, 1--82URL: http://dx.doi.org/10.1007/s006050170032
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• 85 inproceedingsC. Chainais-Hillairet. Entropy method and asymptotic behaviours of finite volume schemesFinite volumes for complex applications. VII. Methods and theoretical aspects77Springer Proc. Math. Stat.Springer, Cham2014, 17--35
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• 86 articleClaireC. Chainais-Hillairet, AnsgarA. Jüngel and StefanS. Schuchnigg. Entropy-dissipative discretization of nonlinear diffusion equations and discrete Beckner inequalitiesModelisation Mathématique et Analyse Numérique5012016, 135-162
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• 87 articleAntoninA. Chambolle, Luca Alberto DavideL. Ferrari and BenoîtB. Merlet. Variational approximation of size-mass energies for k-dimensional currentsESAIM: Control, Optimisation and Calculus of Variations25 (2019)43September 2019, 39
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• 88 articleE. Creusé, S. Nicaise, Z. Tang, Y. Le Menach, N. Nemitz and F. Piriou. Residual-based \it a posteriori estimators for the ${\bf A}-\varphi$ magnetodynamic harmonic formulation of the Maxwell systemMath. Models Methods Appl. Sci.2252012, 1150028-30URL: http://dx.doi.org/10.1142/S021820251150028X
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• 89 articleE. Creusé, S. Nicaise, Z. Tang, Y. Le Menach, N. Nemitz and F. Piriou. Residual-based a posteriori estimators for the $𝐓/$ magnetodynamic harmonic formulation of the Maxwell system'Int. J. Numer. Anal. Model.1022013, 411--429
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• 90 articleE. Creusé, S. Nicaise and E. Verhille. Robust equilibrated a posteriori error estimators for the Reissner-Mindlin systemCalcolo4842011, 307--335URL: http://dx.doi.org/10.1007/s10092-011-0042-0
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• 91 articleB. Després and M. Herda. Correction to: Polynomials with bounds and numerical approximationNumerical Algorithms7712018, 309--311URL: https://doi.org/10.1007/s11075-017-0441-7
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• 92 articleB. Després. Polynomials with bounds and numerical approximationNumerical Algorithms7632017, 829--859URL: https://doi.org/10.1007/s11075-017-0286-0
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• 93 articleDaniele A.D. Di Pietro, AlexandreA. Ern and SimonS. Lemaire. An arbitrary-order and compact-stencil discretization of diffusion on general meshes based on local reconstruction operatorsComput. Methods Appl. Math.1442014, 461--472URL: https://doi.org/10.1515/cmam-2014-0018
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• 94 articleD. A.D. Di Pietro and M. Vohralík. A Review of Recent Advances in Discretization Methods, a Posteriori Error Analysis, and Adaptive Algorithms for Numerical Modeling in GeosciencesOil & Gas Science and Technology-Rev. IFP(online first)June 2014, 1-29
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• 95 articleJeanJ. Dolbeault, ClémentC. Mouhot and ChristianC. Schmeiser. Hypocoercivity for linear kinetic equations conserving massTrans. Amer. Math. Soc.36762015, 3807--3828URL: https://doi.org/10.1090/S0002-9947-2015-06012-7
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• 96 articleV. Dolejš\'\i, A. Ern and M. Vohral\'\ik. A framework for robust a posteriori error control in unsteady nonlinear advection-diffusion problemsSIAM J. Numer. Anal.5122013, 773--793URL: http://dx.doi.org/10.1137/110859282
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• 97 articleJ. Droniou. Finite volume schemes for diffusion equations: introduction to and review of modern methodsMath. Models Methods Appl. Sci.2482014, 1575-1620
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• 98 articleW. E and P. Palffy-Muhoray. Phase separation in incompressible systemsPhys. Rev. E554April 1997, R3844--R3846URL: https://link.aps.org/doi/10.1103/PhysRevE.55.R3844
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• 99 softwareVirginieV. Ehrlacher, ClémentC. Cancès and LaurentL. Monasse. Finite volume scheme for the Stefan--Maxwell model1.0helloJuly 2020,
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• 100 articleE. Emmrich. Two-step BDF time discretisation of nonlinear evolution problems governed by monotone operators with strongly continuous perturbationsComput. Methods Appl. Math.912009, 37--62
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• 101 articleLeslieL. Greengard and June-YubJ.-Y. Lee. Accelerating the nonuniform fast Fourier transformSIAM Rev.4632004, 443--454URL: http://dx.doi.org/10.1137/S003614450343200X
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• 102 articleM. E.M. Hubbard and M. Ricchiuto. Discontinuous upwind residual distribution: a route to unconditional positivity and high order accuracyComput. & Fluids462011, 263--269URL: http://dx.doi.org/10.1016/j.compfluid.2010.12.023
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• 103 articleS. Jin. Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equationsSIAM, J. Sci. Comput.211999, 441-454
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• 104 articleA. Majda and J. Sethian. The derivation and numerical solution of the equations for zero Mach number combustionCombustion Science and Technology421985, 185--205
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• 105 articleF. Otto. The geometry of dissipative evolution equations: the porous medium equationComm. Partial Differential Equations261-22001, 101--174
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• 106 articleM. Ricchiuto and R. Abgrall. Explicit Runge-Kutta residual distribution schemes for time dependent problems: second order caseJ. Comput. Phys.229162010, 5653--5691URL: http://dx.doi.org/10.1016/j.jcp.2010.04.002
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• 107 articleD. Ruppel and E. Sackmann. On defects in different phases of two-dimensional lipid bilayersJ. Phys. France4491983, 1025-1034URL: https://jphys.journaldephysique.org/articles/jphys/abs/1983/09/jphys_1983__44_9_1025_0/jphys_1983__44_9_1025_0.html
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• 108 articleM. Vohral\'\ik. Residual flux-based a posteriori error estimates for finite volume and related locally conservative methodsNumer. Math.11112008, 121--158URL: http://dx.doi.org/10.1007/s00211-008-0168-4
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• 109 articleJ. de Frutos, B. Garc\'\ia-Archilla and J. Novo. A posteriori error estimations for mixed finite-element approximations to the Navier-Stokes equationsJ. Comput. Appl. Math.23662011, 1103--1122URL: http://dx.doi.org/10.1016/j.cam.2011.07.033
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