The project-team SERENA is concerned with numerical methods for environmental problems. The main topics are the conception and analysis of models based on partial differential equations, the study of their precise and efficient numerical approximation, and implementation issues with special concern for reliability and correctness of programs. We are in particular interested in guaranteeing the quality of the overall simulation process.

Within our project, we start from the conception and analysis of models based on partial differential equations (PDEs). We namely address the question of coupling of different models, such as simultaneous fluid flow in a discrete network of two-dimensional fractures and in the surrounding three-dimensional porous medium, or interaction of a (compressible) flow with the surrounding elastic deformable structure. The key physical characteristics need to be captured, whereas existence, uniqueness, and continuous dependence on the data are minimal analytic requirements that we seek to satisfy. We are also interested in localization, approximation, and model reduction.

We consequently design numerical methods for the devised model, while focusing on enabling general polytopal meshes, in particular in response to a high demand from our industrial partners (namely EDF, CEA, and IFP Energies Nouvelles). We in particular promote structure-preserving approaches that mimic at the discrete level the fundamental properties of the underlying PDEs, such as conservation principles and preservation of invariants. We perform numerical analysis in particular in singularly perturbed, unsteady, and nonlinear cases (reaction–diffusion and wave problems, eigenvalue problems, interface problems, variational inequalities, contact problems, degenerate parabolic equations), we apply these methods to challenging problems from fluid and solid mechanics involving large deformations, plasticity, and phase appearance and disappearance, and we develop a comprehensive software implementing them.

We next concentrate an intensive effort on the development and analysis of efficient solvers for the systems of nonlinear algebraic equations that result from the above discretizations. We work on iterative linearization schemes and analysis. We place a particular emphasis on parallelization achieved via the domain decomposition method, including the space-time parallelization for time-dependent problems. This allows the use of different time steps in different parts of the computational domain, particularly useful in our applications where evolution speed varies significantly from one part of the computational domain to another. We have also recently devised novel geometric multigrid solvers with the contraction factor independent of the approximation polynomial degree. The solver itself is adaptively steered at each execution step by an a posteriori error estimate giving a two-sided control of the algebraic error.

The fourth part of our theoretical efforts goes towards assessing the precision of the results obtained at the end of the numerical simulation. Here a key ingredient is the development of rigorous a posteriori estimates that make it possible to estimate in a fully computable way the error between the unknown exact solution and its numerical approximation. Our estimates also allow to distinguish the different components of the overall error, namely the errors coming from modeling, the discretization scheme, the nonlinear (Picard, Newton) solver, and the linear algebraic (domain decomposition, multigrid) solver. A new concept here is that of local stopping criteria, where all the error components are balanced locally within each computational mesh element. This naturally connects all parts of the numerical simulation process and gives rise to novel fully adaptive algorithms. We derive a guaranteed error reduction factor at each adaptive loop iteration in model cases together with cost-optimality in the sense that, up to a generic constant, the smallest possible computational effort to achieve the given accuracy is needed. With patchwise techniques, we also achieve mass balance at each iteration step, a highly demanded feature in most of the target applications.

Finally, we concentrate on the issue of computer implementation of scientific computing programs, noting that precise numerical simulation and guaranteed error estimation are impossible without correct computer implementation. With their increasing complexity, it becomes a major challenge to implement up-to-date scientific computing algorithms using traditional methods and languages. Fortunately, the computer science community has already encountered similar issues, and offers theoretically sound tools for safe and correct programming. We use these tools to design generic solutions for the implementation of the class of scientific computing software the project-team is dealing with. Our focus ranges from high-level programming with OCaml for the precious safety guards provided by its type system and for its ability to encourage functional programming, to proofs of correctness of numerical algorithms and programs, including bounds of the round-off errors, via mechanical proofs with Coq.

[colback=black!5!white] The ultimate objective of the SERENA project-team is to design numerical algorithms that enable to certify the reliability of the overall simulation process and its efficiency with respect to computational resources for the targeted environmental applications.

Via applications with our industrial and environmental partners EDF, CEA, IFP Energies Nouvelles, ANDRA, ITASCA, and BRGM.

Mickael Abbas, Jérôme Bonelle, Nicolas Pignet (EDF R&D) and Alexandre Ern organized the 2023 Edition of the CEA-EdF-Inria summer school on Robust Polyhedral Discretizations for Computational Mechanics (June 26-30, 2023).

The formalization in Coq of simplicial Lagrange finite elements is almost complete. This include the formalizations of the definitions and main properties of monomials, their representation using multi-indices, Lagrange polynomials, the vector space of polynomials of given maximum degree (about 6 kloc). This also includes algebraic complements on the formalization of the definitions and main properties of operators on finite families of any type, the specific cases of abelian monoids (sum), vector spaces (linear combination), and affine spaces (affine combination, barycenter, affine mapping), sub-algebraic structures, and basics of finite dimension linear algebra (about 22 kloc). A new version (2.0) of the opam package will be available soon, and a paper will follow.

We have also contributed to the Coquelicot library by adding the algebraic structure of abelian monoid, which is now the base of the hierarchy of canonical structures of the library.

The code is based on the implementation of the mixed hybrid finite element method as detailed in: An efficient numerical model for incompressible two-phase flow in fractured media Hussein Hoteit, Abbas Firoozabadi, Advances in Water Resources 31, 891–905, 2008. https://doi.org/10.1016/j.advwatres.2008.02.004

The model of fractures and the coupling between the porous flow and the flow in the network of fractures is described in: : Modeling Fractures and Barriers as Interfaces for Flow in Porous Media V. Martin, J. Jaffré, J. E. Roberts, SIAM Journal on Scientific Computing, 2005. https://doi.org/10.1137/S1064827503429363

Validation benchmark test from the publication: Inga Berre, et al., Verification benchmarks for single-phase flow in three-dimensional fractured porous media, Advances in Water Resources, Volume 147, 2021. https://doi.org/10.1016/j.advwatres.2020.103759.

The model proposed as part of the "Multiphase reactive transport" (see Section 8.3) has been archived on Zenodo Reactive Multiphase Flow in Porous Media at the Darcy Scale: a Benchmark proposal. The results obtained by the participants have been made available on Github: Reactive-Multiphase-Benchmark to make it possible for future researchers to compare their results.

In 2022, the authors laid the foundations of a new paradigm for invariant-domain time-stepping applied to hyperbolic problems using high-order Runge–Kutta methods. The key result achieved this year is the extension to implicit-explicit (IMEX) time-stepping and the application to the compressible Navier–Stokes equations, as described in 26. The decisive step-forward is the satisfaction of physical bounds on the density and energy while allowing for a high-order discretization in space and in time. An example of application to the compressible Navier–Stokes equations at Reynolds

Further progress has been accomplished in the development and analysis of hybrid high-order (HHO) methods. Three topics were investigated. First,

In 20, we present a local construction of

In 29, we consider spline/isogeometric analysis discretizations of the Poisson model problem, focusing on high polynomial degrees and strong hierarchical refinements. We derive a posteriori error estimates by equilibrated fluxes, i.e., vector-valued mapped piecewise polynomials lying in the

In 28, we consider nonsmooth partial differential equations associated with a minimization of an energy functional. We adaptively regularize the nonsmooth nonlinearity so as to be able to apply the usual Newton linearization, which is not always possible otherwise. We apply the finite element method as a discretization. We focus on the choice of the regularization parameter and adjust it on the basis of an a posteriori error estimate for the difference of energies of the exact and approximate solutions. We prove guaranteed upper bounds for the energy difference, identify the individual error components, and design an adaptive algorithm with both adaptive regularization and adaptive mesh refinement. Effeciency and robustness of the estimates with respect to the magnitude of the nonlinearity is addressed in 52. Numerical results confirm the theoretical developments, see Figure 3 for an illustration.

See the "News of the Year" about software Skwer (Section 7.1.4).

In 36, we describe the formal definition and proof in Coq of product Lebesgue induction principle provided by an original inductive type for nonnegative measurable functions.

See also the "News of the Year" about software coq-num-analysis (Section 7.1.5).

We have experimented with the domain decomposition preconditioner HPDDM, developped by the Alpines team 55, 56 to solve the linear system obtained from nef-flow-fpm. The improvement over the more classical preconditioners used until then (mainly algebraic multigrid) are significant as can be seen on Figure 4.

Thanks to the gmres solver and the HPDDM preconditioner, we are now able to solve flow problem in large scale fractured porous media that are out of reach with direct solvers like MUMPS Cholesky or with gmres preconditioned by multigrid like BoomerAMG. As example, solving the linear system of size HPDDM, only 2 minutes and 40 iterations with 4096 MPI processes.

Another goal is to simulate the transport by advection of an inert tracer. The transport is described by the conservation of mass and gives rise to an equation with partial derivatives of the first order in which the velocity, computed with the software nef-flow-fpm, is heterogeneous. The discretization in space is performed with a cell-centered finite volume scheme. The discretization in time can be either explicit or implicit.

As part of Alessandra Marelli's internship, we were able to simulate the transport in a network of fractures. The main challenge was the correct handling of the coupling conditions at the fracture intersections. An example is shown in Figure 5. The method is implemented in nef-transport-fpm.

Michel Kern was part of a group with Etienne Ahusborde, Brahim Amaziane (University of Pau), Stephen de Hoop and Denis Voskov (Delft University of Technology) that proposed a benchmark targeted towards the simulation of reactive two-phase flow. Six teams participated in the benchmark. The results showed good agreements between most groups on the simpler test cases, but also that the interaction between complex chemistry and two-phase flow with phase exchanges still remains a challenge for simulation software. The model is presented in 30, while the results are presented in 14, see Figure 6.

As part of the internship of Clément Maradei, we studied a model for the wave equations that includes both a diffusive (first order derivative in time) and a so-called "viscous" term (first order time derivative of the Laplacian). The model has been proposed to represent frequency-dependent attenuation. Thanks to finite element simulations (using FreeFeem++), we were able to compare the respective contributions of the two terms, and use a scaling analysis to better understand the influence of the two parameters. The results have been presented at the 15th FreeFem Days.

Within the PhD Thesis of Romain Mottier, we developed HHO methods to simulate coupled acoustic-elastodynamic waves in geophysical media. One goal is to highlight the role of sedimentary bassins in energy transfer from the bedrock to the atmosphere.

Our work on data assimilation was pursued this year by addressing the heat equation. Our first contribution is on the theoretical side and concerns a Carleman estimate 17. The second contribution deals with the devising and numerical analysis of a high-order method (based on a dG method in time and a hybrid dG method in space) 18.

We design an operator from the infinite-dimensional Sobolev space

In 43, we established the asymptotic optimality of the edge finite element approximation of the time-harmonic Maxwell's equations. This fundamental result, which is the counterpart of a known result concering the Helmholtz equation and conforming finite elements, was still lacking in the litterature. A second novel result, that was also lacking in the literature, concerns the spectral correctness (no spurious eigenvalues) of the dG approximation of Maxwell's equations in first-order form (the result was known for Maxwell's equations in second-order form), thereby confirming numerical observations by various authors made over the last two decades. We proved this result first with constant coefficients 27 and then in the more challenging case of discontinuous coefficients 48.

Diffusive transport in media with discontinuous properties is a challenging problem that arises in many applications. In 39, wefocus on one-dimensional discontinuous media with generalized permeable boundary conditions at the discontinuity interface. The paper presents novel analytical expressions from the method of images to simulate diffusive processes, such as mass or thermal transport. The analytical expressions are used to formulate a generalization of the existing Skew Brownian Motion, HYMLA and Uffink's method, here named as GSBM, GHYMLA and GUM respectively, to handle generic interface conditions. The algorithms rely upon the random walk method and are tested by simulating transport in a bimaterial and in a multilayered medium with piece-wise constant properties. The results indicate that the GUM algorithm provides the best performance in terms of accuracy and computational cost. The methods proposed can be applied for simulation of a wide range of differential problems, like heat transport problem 40.

One important topic has been the development of reduced-order methods to handle variational inequalities such as those encountered when studying contact problems (with friction) in computational mechanics. In 33, we introduce an efficient algorithm to guarantee inf-sup stability for saddle-point problems with parameter-dependent constraints. In 54, we pursued a different, and complementary, approach, where the constraints are taken into account by a nonlinaer Nitsche's method, thereby allowing one to use a primal formulation. Finally, within the PhD Thesis of Abbas Kabalan, we are investigating shape variability within the context of reduced-order models.

In 23, we established optimal decay rates on the best-approximation errors using vector-valued finite elements (of Nédélec or Raviart–Thomas type) for fields with low regularity but having an integrable curl or divergence.

In 25, we derived

These works were performed in collaboration with L. Mascotto.

Prof. Jean-Luc Guermond (Texas A&M University) visited the SERENA team for a comprehensive duration of 15 weeks in 2023 in the framework of his INRIA International Chair. He mainly interacted with Alexandre Ern on invariant-domain preserving high-order time-stepping and on the spectral correctness of discontinuous Galerkin methods for the Maxwell eigenvalue problem, and also with Zhaonan Dong and Zuodong Wang on transport equations with stiff source terms having multiple stable equilibrium points.

EMC2 project on cordis.europa.eu

Molecular simulation has become an instrumental tool in chemistry, condensed matter physics, molecular biology, materials science, and nanosciences. It will allow to propose de novo design of e.g. new drugs or materials provided that the efficiency of underlying software is accelerated by several orders of magnitude.

The ambition of the EMC2 project is to achieve scientific breakthroughs in this field by gathering the expertise of a multidisciplinary community at the interfaces of four disciplines: mathematics, chemistry, physics, and computer science. It is motivated by the twofold observation that, i) building upon our collaborative work, we have recently been able to gain efficiency factors of up to 3 orders of magnitude for polarizable molecular dynamics in solution of multi-million atom systems, but this is not enough since ii) even larger or more complex systems of major practical interest (such as solvated biosystems or molecules with strongly-correlated electrons) are currently mostly intractable in reasonable clock time. The only way to further improve the efficiency of the solvers, while preserving accuracy, is to develop physically and chemically sound models, mathematically certified and numerically efficient algorithms, and implement them in a robust and scalable way on various architectures (from standard academic or industrial clusters to emerging heterogeneous and exascale architectures).

EMC2 has no equivalent in the world: there is nowhere such a critical number of interdisciplinary researchers already collaborating with the required track records to address this challenge. Under the leadership of the 4 PIs, supported by highly recognized teams from three major institutions in the Paris area, EMC2 will develop disruptive methodological approaches and publicly available simulation tools, and apply them to challenging molecular systems. The project will strongly strengthen the local teams and their synergy enabling decisive progress in the field.

The team is part of the recently created GDR HydroGEMM("Hydrogène du sous-sol: étude intégrée de la Genèse... à la Modélisation Mathématique"). One of the thematic axes of the GDR is the mathematical analysis and the numerical simulation hydrogen storage in geological reservoirs.

Alexandre Ern is a member of the Scientific Committee of the European Finite Element Fair.

Mickael Abbas, Jérôme Bonelle, Nicolas Pignet (EDF R&D) and Alexandre Ern organized the 2023 Edition of the CEA-EdF-Inria summer school on Robust Polyhedral Discretizations for Computational Mechanics (June 26-30, 2023).

Alexandre Ern co-organized with Samir Adly (SMAI), R. Herbin (Aix-Marseille University), Nina Aguillon, Xavier Claeys, Bruno Després, Yvon Maday, Ayman Moussa (Sorbonne University) the Month of Applied and Inustrial Mathematics (M2AI) held at IHP on November 2023. Four large-audience lectures were given with the goal to show to undergraduate (and college) students how applied mathematics can (and do) shape our world.

Michel Kern was a member of the organizing committee of the annual meeting of GDR HydroGEMM, held at University of Pau in November 2023.

Pierre Rousselin, Sylvie Boldo (Toccata), François Clément and Micaela Mayero (LIPN) organized the kickoff meeting of the task devoted to the creation of content (math library, exercises, interactive classes) within Inria Challenge LiberAbaci for the teaching of mathematics using Coq.

Martin Vohralík (with Guillaume Enchéry and Ibtihel Ben Gharbia, IFP Energies Nouvelles) organized the regular 1-day workshop Journée contrat cadre IFP Energies Nouvelles – Inria.

Martin Vohralik was a member of the scientific committee of the European Conference on Numerical Mathematics and Advanced Applications ENUMATH 2023.

François Clément served as reviewer for NFM23.

Alexandre Ern is a member of the Editorial Board of SIAM Journal on Scientific Computing, ESAIM Mathematical Modeling and Numerical Analysis, IMA Journal of Numerical Analysis, Journal of Scientific Computing, and Computational Methods in Applied Mathematics.

Martin Vohralík is a member of the editorial boards of Acta Polytechnica, Applications of Mathematics, and Computational Geosciences.

Zhaonan Dong, Alexandre Ern, Michel Kern, Géraldine Pichot, and Martin Vohralík reviewed numerous papers for leading journals in numerical analysis and computational methods in geosciences.

Zhaonan Dong and Géraldine Pichot were invited to organize a mini-tutorial at the SIAM Conference on Mathematical & Computational Issues in the Geosciences 2023, Bergen, Norway, June 2023.

Alexandre Ern gave a plenary lecture at the ECCOMAS Meeting on Modern Finite Element Technologies, Mühlheim an der Ruhr, Germany, August, 2023.

Alexandre Ern gave an invited lecture within the special activity organized by IIT Roorkee, India on Differential equations: analysis, computation and applications.

Géraldine Pichot gave a plenary lecture at the Large-Scale Scientific Computations international conference LSSC23, Sozopol, Bugaria, June 2023.

Martin Vohralík gave an plenary talk at the SIAM Conference on Mathematical and Computational Issues in the Geosciences Bergen, Norway (June 2023), a plenary talk at Congrès international sur l’analyse numérique des EDP, Meknès, Morocco (October 2023), and an invited talk at HOFEIM 2023, Larnaca, Cyprus (May 2023).

Alexandre Ern served within the Administration Board of SMAI and was Vice-President in charge of relations with industry.

Michel Kern is a member of

Michel Kern is a reviewer for the Allocation of Computing Time located at the Juelich Supercomputing Centre in Germany.

François Clément is a member of the Commission des usagers de la rue Barrault (CURB) for the next relocation of the Inria Paris Center.

Michel Kern is the chair of the Comission de Développement Technologique of the Inria Paris Center.

Géraldine Pichot is the president of the Commission des utilisateurs des moyens informatiques de Paris (CUMI Paris).

Géraldine Pichot is a member of the Comité de Suivi Doctoral de Paris (CSD).

Géraldine Pichot is the contact person at Inria Paris for the Agence pour les Mathématiques en Interaction avec l'Entreprise et la Société (AMIES).

Martin Vohralík served in the scientific committee of Summer schools CEA–EDF–INRIA.

Michel Kern gave a presentation on "Modeling and simulation: applications to subsurface water" to a "classe de seconde" at Lycée Lucie Aubrac (Courbevoie) as part of the 1 scientifique, 1 classe : chiche ! project.