2023Activity reportProjectTeamSERENA
RNSR: 201521772E Research center Inria Paris Centre
 In partnership with:Ecole des Ponts ParisTech
 Team name: Simulation for the Environment: Reliable and Efficient Numerical Algorithms
 In collaboration with:Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS)
 Domain:Digital Health, Biology and Earth
 Theme:Earth, Environmental and Energy Sciences
Keywords
Computer Science and Digital Science
 A2.1.3. Objectoriented programming
 A2.1.4. Functional programming
 A2.4.3. Proofs
 A6.1.1. Continuous Modeling (PDE, ODE)
 A6.1.4. Multiscale modeling
 A6.1.5. Multiphysics modeling
 A6.2.1. Numerical analysis of PDE and ODE
 A6.2.5. Numerical Linear Algebra
 A6.2.8. Computational geometry and meshes
 A6.3.1. Inverse problems
 A6.3.4. Model reduction
 A6.3.5. Uncertainty Quantification
Other Research Topics and Application Domains
 B3.1. Sustainable development
 B3.3.1. Earth and subsoil
 B3.4.2. Industrial risks and waste
 B3.4.3. Pollution
 B4.1. Fossile energy production (oil, gas)
 B4.2.1. Fission
 B5.5. Materials
1 Team members, visitors, external collaborators
Research Scientists
 Martin Vohralík [Team leader, INRIA, Senior Researcher, HDR]
 François Clement [INRIA, Researcher]
 Zhaonan Dong [INRIA, Researcher]
 Gregor Gantner [INRIA, ISFP, until Oct 2023]
 JeanCharles Gilbert [INRIA, Emeritus, from Oct 2023]
 JeanLuc Guermond [Texas A&M University, Chair, Inria International Chair, HDR]
 Michel Kern [INRIA, Researcher]
 Geraldine Pichot [INRIA, Researcher]
Faculty Members
 Alexandre Ern [ENPC, Professor, HDR]
 Pierre Rousselin [UNIV PARIS XIII, Professor Delegation, from Sep 2023]
PostDoctoral Fellows
 Akram Beni Hamad [INRIA, PostDoctoral Fellow]
 Andre Harnist [INRIA, PostDoctoral Fellow, until Aug 2023]
PhD Students
 Nicolas Hugot [CEA, from Nov 2023]
 Abbas Kabalan [Safran Tech]
 Clément Maradei [INRIA, from Oct 2023]
 Romain Mottier [ENPC, CEA]
 Houda Mouhcine [Inria Saclay]
 Stefano Piccardo [UPC Barcelona, until Dec 2023]
 Ari Rappaport [INRIA]
 Morgane Steins [CEA, until Dec 2023]
 Zuodong Wang [INRIA]
 Daniel Zegarra Vasquez [INRIA]
Technical Staff
 Sebastien Furic [INRIA, Engineer, until Oct 2023]
 Simon Legrand [Inria, Engineer]
 Raphaël Zanella [Inria, Engineer]
Interns and Apprentices
 Nicolas Hugot [ENSTA, Intern, from Apr 2023 until Oct 2023]
 Clement Maradei [INRIA, Intern, from Feb 2023 until Sep 2023]
 Alessandra Marelli [INRIA, Intern, from Apr 2023 until Sep 2023]
Administrative Assistant
 Derya Gok [INRIA]
Visiting Scientists
 Stefano Bonetti [ECOLE POLYT. MILAN, from May 2023 until Jul 2023]
 Roland Maier [UNIV JENA, from Feb 2023 until Feb 2023]
 Dirk Praetorius [TU WIEN, from Jun 2023 until Jun 2023]
 Fabio Vicini [ECOLE POLYT. TURIN, until Feb 2023]
 Pietro Zanotti [University of Pavia, from Apr 2023 until Feb 2023]
 Lina Zhao [City University of Hong Kong, from Jun 2023 until Jun 2023]
External Collaborators
 Guy Chavent [retired from Inria]
 François Delebecque [retired from Inria]
 Andre Harnist [UTC, from Sep 2023]
 Jérôme Jaffré [retired from Inria, HDR]
 Caroline Japhet [UNIV PARIS XIII]
 Habib Jreige [SciWorks]
 Vincent Martin [UTC]
 Koondanibha Mitra [UNIV EINDHOVEN]
 Jean Roberts [retired from Inria, HDR]
 Pierre Weiss [retired from Inria]
2 Overall objectives
The projectteam 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.
3 Research program
3.1 PDE level
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 twodimensional fractures and in the surrounding threedimensional 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.
3.2 Advanced numerical discretization methods
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 structurepreserving 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.
3.3 Iterative linearization, domain decomposition, and multigrid solvers
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 spacetime parallelization for timedependent 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 twosided control of the algebraic error.
3.4 Reliability by a posteriori error control
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 costoptimality 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.
3.5 Safe and correct programming
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 uptodate 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 projectteam is dealing with. Our focus ranges from highlevel 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 roundoff errors, via mechanical proofs with Coq.
[colback=black!5!white] The ultimate objective of the SERENA projectteam 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.
4 Application domains
4.1 Multiphase flows and transport of contaminants in the subsurface
 fractured and porous media
 flow in largescale discrete fracture networks
 subsurface depollution after chemical leakage
 nuclear waste disposal in deep underground repositories
 geological sequestration of CO2
 production of oil and gas
4.2 Industrial risks in energy production
 structural mechanics (friction, contact, large deformation, plasticity) mainly related to nuclear reactor operation and safety analysis
 Stokes and Navier–Stokes flows related to nuclear reactor operation
 seismic wave propagation for detection and protection
 acoustic wave propagation for non destructive evaluation
 electromagnetism for interfaces between dielectrics and negative metamaterials
5 Social and environmental responsibility
5.1 Impact of research results
Via applications with our industrial and environmental partners EDF, CEA, IFP Energies Nouvelles, ANDRA, ITASCA, and BRGM.
6 Highlights of the year
Mickael Abbas, Jérôme Bonelle, Nicolas Pignet (EDF R&D) and Alexandre Ern organized the 2023 Edition of the CEAEdFInria summer school on Robust Polyhedral Discretizations for Computational Mechanics (June 2630, 2023).
7 New software, platforms, open data
7.1 New software
7.1.1 DiSk++

Name:
Discontinuous Skeletal C++ Library

Keywords:
High order methods, Polyhedral meshes, C++

Scientific Description:
Discontinuous Skeletal methods approximate the solution of boundaryvalue problems by attaching discrete unknowns to mesh faces (hence the term skeletal) while allowing these discrete unknowns to be chosen independently on each mesh face (hence the term discontinuous). Cellbased unknowns, which can be eliminated locally by a Schur complement technique (also known as static condensation), are also used in the formulation. Salient examples of highorder Discontinuous Skeletal methods are Hybridizable Discontinuous Galerkin methods and the recentlydevised Hybrid HighOrder methods. Some major benefits of Discontinuous Skeletal methods are that their construction is dimensionindependent and that they offer the possibility to use general meshes with polytopal cells and nonmatching interfaces. The mathematical flexibility of Discontinuous Skeletal methods can be efficiently replicated in a numerical software: by using generic programming, the DiSk++ library offers an environment to allow a programmer to code mathematical problems in a way completely decoupled from the mesh dimension and the cell shape.

Functional Description:
The software provides a numerical core to discretize partial differential equations arising from the engineering sciences (mechanical, thermal, diffusion). The discretization is based on the "Hybrid highorder" or "Discontinuous Skeletal" methods, which use as principal unknowns polynomials of arbitrary degree on each face of the mesh. An important feature of these methods is that they make it possible to treat general meshes composed of polyhedral cells. The DiSk ++ library, using generic programming techniques, makes it possible to write a code for a mathematical problem independently of the mesh. When a user writes the code for his problem using the basic operations offered by DiSk ++, that code can be executed without modifications on all types of mesh already supported by the library and those that will be added in the future.
 URL:
 Publication:

Author:
Matteo Cicuttin

Contact:
Matteo Cicuttin

Partner:
CERMICS
7.1.2 APSMG

Name:
APosterioriSteered MultiGrid

Keywords:
Finite element modelling, Linear system, A posteriori error estimates, Multigrid methods, Probustness

Scientific Description:
APSMG (aposterioristeered multigrid) is a geometrictype multigrid solver whose execution is steered by the associated a posteriori estimate of the algebraic error. In particular, the descent direction and the levelwise step sizes are adaptively optimized. APSMG corresponds to a Vcycle geometric multigrid with zero pre and solely one postsmoothing step, via blockJacobi (overlapping additive Schwarz/local patchwise problems). Its particularity is that it is robust with respect to the polynomial degree p of the underlying finite element discretization, i.e., APSMG contracts the error on each iteration by a factor that is independent of p. APSMG is the implementation of the solver developed in https://hal.science/hal02070981 and https://hal.science/hal02494538.

Functional Description:
APSMG (aposterioristeered multigrid) is an iterative linear solver implemented in MATLAB. It can treat systems of linear algebraic equations arising from order p conforming finite element discretization of secondorder elliptic diffusion problems. APSMG is a geometrictype multigrid method and uses a hierarchy of nested meshes. It corresponds to a Vcycle geometric multigrid solver with zero pre and one postsmoothing step via blockJacobi (overlapping additive Schwarz/local patchwise problems). A salient feature is the choice of the optimal step size for the descent direction on each mesh level.
 URL:
 Publications:

Contact:
Jan Papez
7.1.3 FEMLAB

Name:
FEMLAB

Keywords:
High order finite elements, Discontinuous Galerkin, Hybrid highorder methods, Adaptive algorithms, Finite element modelling

Functional Description:
FEMLAB is a Matlab library for different classes of FEM code. This library is designed to use a parallel computing toolbox in Matlab to accelerate the time for assembling the linear systems. It has been tested on 48 parallel processors of the HPC nodes. Another critical point is that different FEM codes in this library are designed to support arbitrary order of the basis functions and support the adaptive mesh refinement algorithm.

Release Contributions:
FEMLAB is updated in 2023 to support the adaptive algorithm.
 URL:
 Publications:

Contact:
Zhaonan Dong
7.1.4 Skwer

Keywords:
Differential equations, Stateoriented simulation

Scientific Description:
Unlike classical approaches which discretize time a priori to determine the state, the StateOriented Simulation (SOS) method discretizes the state to determine durations, thus following ideas from the Quantized State Systems (QSS) methods. The aim is to give rigorous interpretation of idealized or hybrid physical models, and of cosimulation. The aspect that most distinguishes us from other approaches is that we do not make use of differentialalgebraic equations (DAEs). We rather solve more general multiscale numerical stiffness problems based on nonstandard ODE formulations.

Functional Description:
Skwer aims at solving differential equations inherent to 0D physical modeling. This includes in particular standard ODEs but also differential equations with conditionals and "idealized behavior" resulting from passing to the limit over some parameters.

News of the Year:
We have developed the necessary machinery to desynchronize elementary integrators making the approach fully asynchronous event in dense cases (contrary to the QSS approach, which requires synchronization of internal states in case of direct variableequation dependency between integrators). We have finalized system rewriting aspects (various enhancement of data structures as well as algorithms). We have developed a variable order scheme where each state variable may have its own order (independent of others) thanks to the use of a formulabased technique to produce guaranteed a priori error bounds for the individual approximations. We have finally dropped the Maclaurin series code in favor of a more specialized (and lighter) approach based on direct generation of exponential terms. We have tested the final algorithm over various systems of differential equations to validate the approach.
 URL:
 Publication:

Contact:
Sebastien Furic

Participants:
Sebastien Furic, François Clement, Geraldine Pichot
7.1.5 coqnumanalysis

Name:
Numerical analysis Coq library

Keywords:
Coq, Numerical analysis, Real analysis

Scientific Description:
These Coq developments are based on the Coquelicot library for real analysis. Version 1.0 includes the formalization and proof of: (1) the LaxMilgram theorem, including results from linear algebra, geometry, functional analysis and Hilbert spaces, (2) the Lebesgue integral, including large parts of the measure theory,the building of the Lebesgue measure on real numbers, integration of nonnegative measurable functions with the Beppo Levi (monotone convergence) theorem, Fatou's lemma, the Tonelli theorem, and the Bochner integral with the dominated convergence theorem.

Functional Description:
Formal developments and proofs in Coq of numerical analysis problems. The current longterm goal is to formally prove parts of a C++ library implementing the Finite Element Method.

News of the Year:
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 multiindices, 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), subalgebraic 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.
 URL:
 Publications:

Contact:
Sylvie Boldo

Participants:
Sylvie Boldo, François Clement, Micaela Mayero, Vincent Martin, Stéphane Aubry, Florian Faissole, Houda Mouhcine, Louise Leclerc

Partners:
LIPN (Laboratoire d'Informatique de l'Université Paris Nord), LMAC (Laboratoire de Mathématiques Appliquées de Compiègne)
7.1.6 MODFRAC

Name:
MODFRAC

Keywords:
Meshing, Fracture network, Ellipses, Polygons, Mesher, Mesh

Scientific Description:
The meshing methodology is based on a combined frontalDelaunay approach in a Riemannian context.

Functional Description:
The MODFRAC software automatically builds meshes of fracture networks. As an input, it takes a DFN (Discrete Fracture Network) geometric model consisting of ellipses or polygons that have been randomly generated in the tridimensional space while following experimental statistics. It completes this model by first calculating the intersections between fractures, that are straight segments. On each fracture, it computes in turn the intersections between these straight segments, subdividing them into subsegments. It then creates a conforming set of these subsegments, and selects the necessary fractures using a graph structure. It transmits this information to an “indirect” surface mesher, where the tridimensional mesh results from the construction of planar meshes of the parametric domains.

News of the Year:
APP deposit, january 2023. Addition of OpenMP parallelism.
 Publications:

Contact:
Geraldine Pichot

Participants:
Patrick Laug, Houman Borouchaki, Geraldine Pichot

Partner:
Université de Technologie de Troyes
7.1.7 nefflowfpm

Keywords:
2D, 3D, Porous media, Fracture network, Geophysical flows

Scientific Description:
The code is based on the implementation of the mixed hybrid finite element method as detailed in: An efficient numerical model for incompressible twophase 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 singlephase flow in threedimensional fractured porous media, Advances in Water Resources, Volume 147, 2021. https://doi.org/10.1016/j.advwatres.2020.103759.

Functional Description:
nefflowfpm is a Matlab code to simulate flows in fractured porous media with the mixedhybrid finite element methods (RT0).

Release Contributions:
Implementation of the mixed hybrid method for 3D porous flows, Discrete fracture Networks (DFN) flows and the coupling between DFN and porous flows.

News of the Year:
Call metis to perform the mesh partitionning. Generate data per subdomain for HPDDM solver (Neumann matrices, local sizes and indices, local second member). Simulations with PETSC/HPDDM, the largest networks contains 378k fractures.
 URL:

Contact:
Geraldine Pichot

Participants:
Geraldine Pichot, Daniel Zegarra Vasquez, Michel Kern, Raphael Zanella
7.1.8 neftransportfpm

Keywords:
3D, Porous media, Incompressible flows, Transport model

Scientific Description:
The discretization in space is performed with a cellcentered finite volume scheme. The discretization in time can be either explicit or implicit.

Functional Description:
neftransportfpm is a Matlab code for simulating transport by advection diffusion in porousfractured media.

News of the Year:
Add transport in a network of fractures with the proper handling of the coupling conditions at the fractures intersections.

Contact:
Geraldine Pichot

Participants:
Geraldine Pichot, Michel Kern, Daniel Zegarra Vasquez, Alessandra Marelli, Dania Khiralla
7.1.9 ParaCirce

Name:
Parallel Circulant Embedding

Keywords:
2D, 3D, Hydrogeology, Gaussian random fields, MPI

Scientific Description:
ParaCirce implements the algorithm proposed by [C. R. Dietrich and G. N. Newsam. A fast and exact method for multidimensional gaussian stochastic simulations. Water Resources Research, 29(8):28612869, 1993] as well as an algorithm to accelerate the padding estimation [Pichot et al. SMAI Journal of Computational Mathematics, 8, pp.21, 2022].

Functional Description:
ParaCirce implements a parallel Circulant Embedding method for the generation in parallel of 2D or 3D Gaussian Random Fields (second order stationary).

Release Contributions:
A new GRF type has been introduced to wrap the std::vector initially returned from generate(). It contains all the characteristics of the GRF. Add API for the FFTW to select the planning strategy.

News of the Year:
A new GRF type has been introduced to wrap the std::vector initially returned from generate(). It contains all the characteristics of the GRF. Add API for the FFTW to select the planning strategy.
 URL:
 Publication:

Contact:
Geraldine Pichot

Participants:
Geraldine Pichot, Simon Legrand
7.1.10 Pruners

Name:
Pruners

Keywords:
Combinatorics, Parameter studies, Automation

Functional Description:
Pruners is a language aimed at automating parameter studies. It allows the specification of parameter combinations, and make them available via environment variables. Those can then be used by any specified command as input parameters.

Release Contributions:
 Parameter space specification  Launching commands with each combination as parameter  File system interaction with read/write functions and Json format

News of the Year:
Open source licence GPL and opening of the repository. Add skip and keep options to select particular combinations of parameters. Add configure option to replace pruners script variables into a templated file. Add asynchronous execution of combinations.
 URL:

Authors:
Thierry Martinez, Simon Legrand, Geraldine Pichot

Contact:
Thierry Martinez
7.2 Open data
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: ReactiveMultiphaseBenchmark to make it possible for future researchers to compare their results.
8 New results
8.1 Research axis 1: Advanced numerical discretizations and solvers
Participants: Zhaonan Dong, Alexandre Ern, JeanLuc Guermond, Michel Kern, Stefano Piccardo, Morgane Steins, Martin Vohralík, Zuodong Wang.
Invariantdomain timestepping for compressible flows
Participants: Alexandre Ern, JeanLuc Guermond, Zuodong Wang.
In 2022, the authors laid the foundations of a new paradigm for invariantdomain timestepping applied to hyperbolic problems using highorder Runge–Kutta methods. The key result achieved this year is the extension to implicitexplicit (IMEX) timestepping and the application to the compressible Navier–Stokes equations, as described in 26. The decisive stepforward is the satisfaction of physical bounds on the density and energy while allowing for a highorder discretization in space and in time. An example of application to the compressible Navier–Stokes equations at Reynolds $Re=1000$ is displayed in Figure 1. This is a very challenging problem owing to the interactions between shocks and walls and the development of multiscale vortical structures. Moreover, in 49, we considered a scalar conservation law with a stiff source term having multiple equilibrium points. For this quite challenging situation, we proposed a scheme that can be asymptoticpreserving.
Polytopal discretization methods
Participants: Zhaonan Dong, Alexandre Ern, JeanLuc Guermond, Stefano Piccardo, Morgane Steins, Zuodong Wang.
Further progress has been accomplished in the development and analysis of hybrid highorder (HHO) methods. Three topics were investigated. First, ${C}^{0}$HHO methods for the biharmonic problem in 22 leading to a competitive method in terms of error vs. computational effort with respect to estabished methods such as the ${C}^{0}$interior penalty discontinuous Galerkin method. Second, within the framework of the PhD Thesis of Morgane Steins (38,defended this year), HHO methods for the wave equation using a leapfrong scheme for time discretization were studied. The contributions include a convergence analysis 50 and a timeexplicit marching scheme 35. Finally, HHO methods were used to study surface tension effects between two immiscible Stokes fluids within the PhD Thesis of Stefano Piccardo (37, defended this year) 34.
8.2 Research axis 2: A posteriori error control, adaptivity, and safe and correct programming
Participants: François Clément, Alexandre Ern, Sébastien Furic, Gregor Gantner, André Harnist, Houda Mouhcine, Ari Rappaport, Martin Vohralík.
Error control with quality uniform in polynomial degree for the curl–curl (simplified Maxwell) problem
Participants: Martin Vohralík.
In 20, we present a local construction of $\mathbf{H}\left(\mathrm{curl}\right)$conforming piecewise polynomials satisfying a prescribed curl constraint. We start from a piecewise polynomial not contained in the $\mathbf{H}\left(\mathrm{curl}\right)$ space but satisfying a suitable orthogonality property. The procedure employs minimizations in vertex patches and the outcome is, up to a generic constant independent of the underlying polynomial degree, as accurate as the bestapproximations over the entire local versions of $\mathbf{H}\left(\mathrm{curl}\right)$. This allows to design guaranteed, fully computable, constantfree, and polynomialdegreerobust a posteriori error estimates of the Prager–Synge type for Nédélec's finite element approximations of the curl–curl problem. A divergencefree decomposition of a divergencefree $\mathbf{H}\left(\mathrm{div}\right)$conforming piecewise polynomial, relying on overconstrained minimizations in Raviart–Thomas' spaces, is the key ingredient. Numerical results confirm the theoretical developments, see Figure 2 for an illustration.
Error control with quality uniform in spline degree for isogeometric analysis
Participants: Gregor Gantner, Martin Vohralík.
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., vectorvalued mapped piecewise polynomials lying in the $\mathbf{H}\left(\mathrm{div}\right)$ space which appropriately approximate the desired divergence constraint. Our estimates are constantfree in the leading term, locally efficient, and robust with respect to the polynomial degree. They are also robust with respect to the number of hanging nodes arising in adaptive mesh refinement employing hierarchical Bsplines. Two partitions of unity are designed, one with larger supports corresponding to the mapped splines, and one with small supports corresponding to mapped piecewise multilinear finite element hat basis functions. The equilibration is only performed on the small supports, avoiding the higher computational price of equilibration on the large supports or even the solution of a global system. Thus, the derived estimates are also as inexpensive as possible. An abstract framework for such a setting is developed, whose application to a specific situation only requests a verification of a few clearly identified assumptions. Numerical experiments illustrate the theoretical developments and even indicate, though not rigorougsly proved, robustness with respect to the smoothness of the splines.
Adaptive regularization, discretization, and linearization for nonsmooth problems
Participants: André Harnist, Ari Rappaport, Martin Vohralík.
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.
Functional software
Participants: Sébastien Furic.
See the "News of the Year" about software Skwer (Section 7.1.4).
Coq formalizations
Participants: François Clément, Houda Mouhcine.
In 36, we describe the formal definition and proof in Coq of product $\sigma $algebras, product measures and their uniqueness, the construction of iterated integrals, up to Tonelli's theorem. We also advertise the Lebesgue induction principle provided by an original inductive type for nonnegative measurable functions.
See also the "News of the Year" about software coqnumanalysis (Section 7.1.5).
8.3 Research axis 3: Applications to environment and energy
Participants: Alexandre Ern, Michel Kern, Simon Legrand, Clément Maradei, Alessandra Marelli, Romain Mottier, Géraldine Pichot, Martin Vohralík, Raphaël Zanella, Daniel Zegarra Vasquez.
Flow through fractured and fractured porous media
Participants: Michel Kern, Simon Legrand, Alessandra Marelli, Géraldine Pichot, Martin Vohralík, Raphaël Zanella, Daniel Zegarra Vasquez.
We have experimented with the domain decomposition preconditioner HPDDM, developped by the Alpines team 55, 56 to solve the linear system obtained from nefflowfpm. 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 $1.41\times {10}^{8}$ for a network containing 378k fractures takes in parallel, with gmres preconditioned with 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 nefflowfpm, is heterogeneous. The discretization in space is performed with a cellcentered 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 neftransportfpm.
Multiphase reactive flow
Participants: Michel Kern.
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 twophase 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 twophase 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.
Wave propagation in geophysical media
Participants: Alexandre Ern, Michel Kern, Clément Maradei, Romain Mottier.
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 socalled "viscous" term (first order time derivative of the Laplacian). The model has been proposed to represent frequencydependent 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 acousticelastodynamic waves in geophysical media. One goal is to highlight the role of sedimentary bassins in energy transfer from the bedrock to the atmosphere.
Data assimilation
Participants: Alexandre Ern.
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 highorder method (based on a dG method in time and a hybrid dG method in space) 18.
8.4 Research axis 4: PDE and numerical analysis foundations
Participants: Zhaonan Dong, Alexandre Ern, JeanLuc Guermond, Géraldine Pichot, Martin Vohralík.
A stable local commuting projector and optimal $hp$ approximation estimates in $\mathbf{H}\left(\mathrm{curl}\right)$
Participants: Théophile ChaumontFrelet, Martin Vohralík.
We design an operator from the infinitedimensional Sobolev space $\mathbf{H}\left(\mathrm{curl}\right)$ to its finitedimensional subspace formed by the Nédélec piecewise polynomials on a tetrahedral mesh that has the following properties: 1) it is defined over the entire $\mathbf{H}\left(\mathrm{curl}\right)$, including boundary conditions imposed on a part of the boundary; 2) it is defined locally in a neighborhood of each mesh element; 3) it is based on simple piecewise polynomial projections; 4) it is stable in the ${\mathbf{L}}^{2}$norm, up to data oscillation; 5) it has optimal (localbest) approximation properties; 6) it satisfies the commuting property with its sibling operator on $\mathbf{H}\left(\mathrm{div}\right)$; 7) it is a projector, i.e., it leaves intact objects that are already in the Nédélec piecewise polynomial space. This operator can be used in various parts of numerical analysis related to the $\mathbf{H}\left(\mathrm{curl}\right)$ space. We in particular employ it here to establish the two following results: i) equivalence of globalbest, tangentialtrace and curlconstrained, and localbest, unconstrained approximations in $\mathbf{H}\left(\mathrm{curl}\right)$ including data oscillation terms; and ii) fully $h$ and $p$ (meshsize and polynomialdegree) optimal approximation bounds valid under the minimal Sobolev regularity only requested elementwise.
Some novel results concerning Maxwell's equations
Participants: Alexandre Ern, JeanLuc Guermond.
In 43, we established the asymptotic optimality of the edge finite element approximation of the timeharmonic 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 firstorder form (the result was known for Maxwell's equations in secondorder 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.
Solutions to 1D advectiondiffusion problems with discontinuous coefficients
Participants: Géraldine Pichot.
Diffusive transport in media with discontinuous properties is a challenging problem that arises in many applications. In 39, wefocus on onedimensional 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 piecewise 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.
Modelorder reduction
Participants: Alexandre Ern, Abbas Kabalan.
One important topic has been the development of reducedorder 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 infsup stability for saddlepoint problems with parameterdependent 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 reducedorder models.
Bestapproximation errors for fields with low regularity
Participants: Zhaonan Dong, Alexandre Ern, JeanLuc Guermond.
In 23, we established optimal decay rates on the bestapproximation errors using vectorvalued finite elements (of Nédélec or Raviart–Thomas type) for fields with low regularity but having an integrable curl or divergence.
$hp$optimal error estimates of dG methods
Participants: Zhaonan Dong.
In 25, we derived $hp$optimal error estimates for dG methods for the biharmonic problem with homogeneous essential boundary conditions, which removed the $1.5$ suboptimal rate in term of $p$ in the classical error analysis of dG methods. The main ingredient in the analysis is the construction of a global ${H}^{2}$ piecewise polynomial approximants with $hp$optimal approximation properties over the meshes. Moreover recently, we derived $hp$optimal error estimates for the upwind dG method when approximating solutions to firstorder hyperbolic problems with constant convection fields in the ${L}_{2}$ and DG norms in 47. The main novelty in the analysis are novel $hp$optimal approximation properties of the special projector introduced in [Cockburn, Dong, Guzman, SINUM, 2008].
These works were performed in collaboration with L. Mascotto.
9 Bilateral contracts and grants with industry
9.1 Bilateral contracts with industry
Participants: Alexandre Ern, Martin Vohralík.
 Twopart contract with CEA accompanying the PhD thesis of Nicolas Hugot.
 Twopart contract with Safran Tech accompanying the PhD thesis of Abbas Kabalan (cosupervised with V. Ehrlacher).
 Twopart contract with CEA accompanying the PhD thesis of Romain Mottier.
 Twopart contract with ANDRA accompanying the PhD thesis of Ari Rappaport.
 Twopart contract with CEA accompanying the PhD thesis of Morgane Steins.
10 Partnerships and cooperations
10.1 International research visitors
10.1.1 Visits of international scientists
Inria International Chair
Prof. JeanLuc 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 invariantdomain preserving highorder timestepping 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.
Other international visits to the team
Dirk Praetorius

Status
researcher
 Institution of origin:

Country:
Austria

Dates:
june 2023

Context of the visit:
Dirk Praetorius visited us in June 2023. He is a worldleading expert on adaptive mesh refinement, adaptive solvers, and proofs of convergence and optimality. He mainly interacted with Martin Vohralík. He has by now also applied for the Inria International Chair.

Mobility program/type of mobility:
research stay
Fabio Vicini

Status
researcher
 Institution of origin:

Country:
Italy

Dates:
december 2022  january 2023

Context of the visit:
Fabio works in the GEOSCORE group of Politecnico di Torino. GEOSCORE and SERENA have a common research topic that is the development of efficient and robust numerical methods to solve largescale subsurface flows. To this end, during the past years, our respective teams have developed different meshing and numerical strategies. During Fabio's stay, he mainly interacted with Géraldine Pichot to compare these different approaches: matching vs nonmatching meshes, Virtual Element Method vs Hybrid High Order method, direct vs iterative solvers, adaptive mesh refinement strategies based on a posteriori error estimates.

Mobility program/type of mobility:
research stay
Lina Zhao

Status
Assistant Professor

Institution of origin:
City University of Hong Kong

Country:
China

Dates:
June 2023  June 2023

Context of the visit:
Dr. Lina Zhao (City University of Hong Kong) visited the SERENA team for 4 weeks in June 2023. She mainly interacted with Zhaonan Dong and Alexandre Ern on minimum regularity of Staggered DG methods for the flow problems, and a posteriori error analysis for the coupled problems.

Mobility program/type of mobility:
research stay
10.1.2 Visits to international teams
Research stays abroad
Martin Vohralík
 Visited institution: University of Texas at Austin
 Country: USA
 Dates: 9  25 Jan 2023
 Context of the visit: scientific collaboration with Prof. Leszek Demkowicz on commuting projects

Mobility program/type of mobility:
research stay
10.2 European initiatives
10.2.1 H2020 projects
EMC2
EMC2 project on cordis.europa.eu

Title:
Extremescale Mathematicallybased Computational Chemistry

Duration:
From September 1, 2019 to February 28, 2026

Partners:
 INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET AUTOMATIQUE (INRIA), France
 ECOLE NATIONALE DES PONTS ET CHAUSSEES (ENPC), France
 CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE CNRS (CNRS), France
 SORBONNE UNIVERSITE, France

Inria contact:
Laura GRIGORI (Alpines)
 Coordinator:

Summary:
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 multimillion 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 stronglycorrelated 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.
10.3 National initiatives
Participants: Michel Kern.
The team is part of the recently created GDR HydroGEMM("Hydrogène du soussol: 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.
11 Dissemination
Participants: François Clément, Zhaonan Dong, Alexandre Ern, Gregor Gantner, JeanLuc Guermond, Michel Kern, Martin Vohralík.
11.1 Promoting scientific activities
11.1.1 Scientific events: organisation
Member of the organizing committees
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 CEAEdFInria summer school on Robust Polyhedral Discretizations for Computational Mechanics (June 2630, 2023).
Alexandre Ern coorganized with Samir Adly (SMAI), R. Herbin (AixMarseille 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 largeaudience 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 1day workshop Journée contrat cadre IFP Energies Nouvelles – Inria.
11.1.2 Scientific events: selection
Member of the conference program committees
Martin Vohralik was a member of the scientific committee of the European Conference on Numerical Mathematics and Advanced Applications ENUMATH 2023.
Reviewer
François Clément served as reviewer for NFM23.
11.1.3 Journal
Member of the editorial boards
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.
Reviewer  reviewing activities
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.
11.1.4 Invited talks
Zhaonan Dong and Géraldine Pichot were invited to organize a minitutorial 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 LargeScale 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).
11.1.5 Leadership within the scientific community
Alexandre Ern served within the Administration Board of SMAI and was VicePresident in charge of relations with industry.
Michel Kern is a member of
 the Scientific Board of ORAP, Organisation Associative du Parallélisme;
 the board of École Doctorale Galilée at University Sorbonne ParisNord;
 the steering committee of GDR HydroGEMM
 Martin Vohralík served as the president/member of the scientific committee of Summer schools CEA–EDF–INRIA.
 Martin Vohralík served as a member of the scientific board of the IFP Energies Nouvelles – Inria joint strategic partnership laboratory.
11.1.6 Scientific expertise
Michel Kern is a reviewer for the Allocation of Computing Time located at the Juelich Supercomputing Centre in Germany.
11.1.7 Research administration
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).
11.2 Teaching  Supervision  Juries
11.2.1 Teaching
 Master : Alexandre Ern, Discontinuous Galerkin methods, 20h, M2, Sorbonne University, France.
 Master: Alexandre Ern, Finite Elements, 15h, M1, ENPC, France.
 Master: Alexandre Ern, Hyperbolic equations, 6h, M2, Sorbonne University, France.
 Master: Michel Kern, Models and numerical methods for subsurface flow, 30h, M2, Université Paris Saclay, France.
 Master: Michel Kern, Advanced numerical analysis, 30h, M1, Institut Galilée, Université ParisNord, France.
 Master: Martin Vohralík, Advanced finite elements, 21h, M1, ENSTA (Ecole nationale supérieure de techniques avancées), Paris, France.
11.2.2 Supervision
 PhD defended: Stefano Piccardo, Simulation of twofluid immiscible Stokes flows using hybrid nonconforming methods and geometrically unfitted meshes, 04 December 2023, Alexandre Ern and Antonio Huerta (UPC Barcelone), 37.
 PhD defended: Morgane Steins, An explicit hybrid highorder method for structural dynamics, 05 December 2023, Alexandre Ern and Olivier Jamond (CEA), 38.
 PhD in progress: Nicolas Hugot, A posteriori error estimates for the wave equation, started November 2023, Martin Vohralík.
 PhD in progress: Abbas Kabalan, Model order reduction for nonparametrized geometrical variability, started October 2022, Virginie Ehrlacher (Matherials) and Alexandre Ern.
 PhD in progress: Clément Maradei, Parallel adaptive $hp$FEM, started October 2023, Zhaonan Dong and Martin Vohralík.
 PhD in progress: Romain Mottier, Unfitted hybrid highorder methods for geophysical wave propagation, started October 2021, Alexandre Ern and Laurent Guillot (CEA).
 PhD in progress: Houda Mouhcine, Formal proofs in applied mathematics: verification of a generator for quadrature formulas, started October 2021, Sylvie Boldo (Toccata), François Clément, and Micaela Mayero (LIPN).
 PhD in progress: Ari Rappaport, A posteriori error estimates and adaptivity in numerical approximation of PDEs: regularization, linearization, discretization, and floating point precision, started January 2021, Martin Vohralík.
 PhD in progress: Daniel Zegarra Vasquez, Highperformance simulation of singlephase flows in a fractured porous medium, started October 2021, Géraldine Pichot, Michel Kern, and Martin Vohralík.
 PhD in progress: Zuodong Wang, Finite element methods for hyperbolic and degenerate parabolic problems, started October 2021, Zhaonan Dong and Alexandre Ern.
 Internship: Nicolas Hugot (ENSTA), Flux reconstruction for the wave equation, AprilSeptember 2023, Martin Vohralík.
 Internship: Clément Maradei (Sup Galilée), Finite element simulation of the viscous wave equation, FebruaryMarch 2023, Michel Kern.
 Internship: Clément Maradei (Sup Galilée), Parallel adaptive $hp$FEM, AprilSeptember 2023, Zhaonan Dong and Martin Vohralík.
 Internship: Alessandra Marelli (Politecnico di Milano), Simulation of advective transport in fracture networks, AprilSeptember 2023, Géraldine Pichot and Michel Kern.
11.2.3 Juries
 Alexandre Ern was external reviewer for the PhD of Matthieu Barré (IPP), Simon Le Berre (CEA Cadarache and Mines ParisTech), and Kenneth Assogba (IPP) and committee member for the HDR of Laurent Monasse (Cote d'Azur University) and for the PhD of Julien Moatti (Lille University).
 Martin Vohralík was a referee and committee president for the PhD thesis of Hussein Albazzal (Université de Bourgogne FrancheComté), committee president for the PhD thesis of Georges Seeman (Université SaintJoseph de Beyrouth), and committee member for the PhD thesis of Sarah Perez (Université de Pau et des Pays Adour) and Yipeng Wang (Sorbonne Université).
11.3 Popularization
11.3.1 Internal or external Inria responsibilities
Martin Vohralík served in the scientific committee of Summer schools CEA–EDF–INRIA.
11.3.2 Interventions
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.
12 Scientific production
12.1 Major publications
 1 articleA posteriori error estimates and stopping criteria for spacetime domain decomposition for twophase flow between different rock types.SMAI Journal of Computational Mathematics5December 2019, 195227HALDOI
 2 articlePreconditioning a coupled model for reactive transport in porous media.International Journal of Numerical Analysis and Modeling1612019, 1848HAL
 3 inproceedingsA Coq formal proof of the Lax–Milgram theorem.6th ACM SIGPLAN Conference on Certified Programs and ProofsParis, FranceJanuary 2017HALDOI
 4 articleTrusting computations: A mechanized proof from partial differential equations to actual program.Computers and Mathematics with Applications683August 2014, 325352HALDOI
 5 articleGuaranteed and robust a posteriori bounds for Laplace eigenvalues and eigenvectors: conforming approximations.SIAM Journal on Numerical Analysis555September 2017, 22282254HALDOI

6
article
$hp$ Version discontinuous Galerkin methods on essentially arbitrarilyshaped elements.Mathematics of Computation91333January 2022, 135HALDOI  7 articleA hybrid highorder lockingfree method for linear elasticity on general meshes.Comput. Methods Appl. Mech. Engrg.2832015, 121URL: http://dx.doi.org/10.1016/j.cma.2014.09.009DOI

8
articleEquivalence of localand globalbest approximations, a simple stable local commuting projector, and optimal
$hp$ approximation estimates in$H$ (div).IMA Journal of Numerical Analysis422April 2022, 10231049HALDOI  9 articleFinite element quasiinterpolation and best approximation.ESAIM Math. Model. Numer. Anal.5142017, 13671385URL: https://doi.org/10.1051/m2an/2016066
 10 miscHybrid highorder methods for flow simulations in extremely large discrete fracture networks.November 2022HAL
 11 articlePolynomialdegreerobust a posteriori estimates in a unified setting for conforming, nonconforming, discontinuous Galerkin, and mixed discretizations.SIAM Journal on Numerical Analysis532April 2015, 10581081HALDOI
 12 articleStable broken H1 and H(div) polynomial extensions for polynomialdegreerobust potential and flux reconstruction in three space dimensions.Mathematics of Computation89322March 2020, 551594HALDOI
 13 articleSpacetime domain decomposition methods for diffusion problems in mixed formulations.SIAM J. Numer. Anal.5162013, 35323559URL: http://dx.doi.org/10.1137/130914401DOI
12.2 Publications of the year
International journals
 14 articleA benchmark study on reactive twophase flow in porous media: Part II results and discussion.Computational Geosciences2023HALback to text
 15 articleAdaptive inexact smoothing Newton method for a nonconforming discretization of a variational inequality.Computers & Mathematics with Applications1332023, 1229HALDOI
 16 articleSemismooth and smoothing Newton methods for nonlinear systems with complementarity constraints: Adaptivity and inexact resolution.Journal of Computational and Applied Mathematics4202023, 114765HALDOI
 17 articleA stability estimate for data assimilation subject to the heat equation with initial datum.Comptes Rendus. Mathématique2024HALback to text
 18 articleThe unique continuation problem for the heat equation discretized with a highorder spacetime nonconforming method.SIAM Journal on Numerical Analysis6152023, 25342557HALback to text
 19 articleA posteriori error estimates for discontinuous Galerkin methods on polygonal and polyhedral meshes.SIAM Journal on Numerical Analysis6152023, 23522380HALDOI
 20 articleprobust equilibrated flux reconstruction in H(curl) based on local minimizations. Application to a posteriori analysis of the curlcurl problem.SIAM Journal on Numerical Analysis614July 2023, 17831818HALDOIback to textback to text

21
articleGuaranteed contraction of adaptive inexact
$hp$ refinement strategies with realistic stopping criteria.ESAIM: Mathematical Modelling and Numerical Analysis571February 2023, 329  366HALDOI 
22
article
${C}^{0}$ hybrid highorder methods for biharmonic problems.IMA Journal of Numerical Analysis441February 2024, 24–57HALDOIback to text  23 articleLocal decay rates of bestapproximation errors using vectorvalued finite elements for fields with low regularity and integrable curl or divergence.Comptes Rendus. Mathématique361May 2023, 723736HALDOIback to text
 24 articleAn improved highorder method for elliptic multiscale problems.SIAM Journal on Numerical Analysis6142023, 19181937HALDOI
 25 articlehpoptimal interior penalty discontinuous Galerkin methods for the biharmonic problem.Journal of Scientific Computing9630May 2023HALDOIback to text
 26 articleInvariantdomain preserving highorder time stepping: II. IMEX schemes *.SIAM Journal on Scientific Computing4552023, A2511A2538HALback to textback to text
 27 articleThe discontinuous Galerkin approximation of the graddiv and curlcurl operators in firstorder form is involutionpreserving and spectrally correct.SIAM Journal on Numerical Analysis2024HALback to text
 28 articleAdaptive regularization, discretization, and linearization for nonsmooth problems based on primaldual gap estimators.Computer Methods in Applied Mechanics and Engineering418May 2023, 116558HALDOIback to textback to text
 29 articleInexpensive polynomialdegreerobust equilibrated flux a posteriori estimates for isogeometric analysis.Mathematical Models and Methods in Applied Sciences2023, 146HALDOIback to text
 30 articleA benchmark study on reactive twophase flow in porous media: Part I model description.Computational Geosciences2023HALback to text
 31 articleA spacetime multiscale mortar mixed finite element method for parabolic equations.SIAM Journal on Numerical Analysis612April 2023, 675  706HALDOI
 32 articleA posteriori error estimates for the Richards equation.Mathematics of ComputationDecember 2023HAL
 33 articleStable model reduction for linear variational inequalities with parameterdependent constraints.ESAIM: Mathematical Modelling and Numerical Analysis5712023, 167189HALback to text
 34 articleSurface tension effects between two immiscible Stokes fluids: a computational study using unfitted hybrid highorder methods and a levelset scheme.SMAI Journal of Computational Mathematics92023, 257283HALback to text
 35 articleTimeexplicit Hybrid HighOrder method for the nonlinear acoustic wave equation.ESAIM: Mathematical Modelling and Numerical Analysis5752023, 29773006HALback to text
International peerreviewed conferences
 36 inproceedingsA Coq Formalization of Lebesgue Induction Principle and Tonelli's Theorem.Proceedings of the 25th International Symposium on Formal Methods25th International Symposium on Formal Methods (FM 2023)14000Lecture Notes in Computer ScienceLübeck, GermanyMarch 2023, 3955HALDOIback to text
Doctoral dissertations and habilitation theses
 37 thesisSimulation of twofluid immiscible Stokes flows using hybrid nonconforming methods and geometrically unfitted meshes.Ecole des Ponts ParisTech; Universitat Politècnica de Catalunya, Barcelona (Spain)December 2023HALback to textback to text
 38 thesisAn explicit hybrid highorder method for structural dynamics.ENPC  École des Ponts ParisTechDecember 2023HALback to textback to text
Reports & preprints
 39 miscModeling diffusion in discontinuous media under generalized interface conditions: theory and algorithms.July 2023HALback to text
 40 miscRandom walk modeling of conductive heat transport in discontinuous media.July 2023HALback to text
 41 reportLebesgue Induction and Tonelli's Theorem in Coq.RR9457Institut National de Recherche en Informatique et en Automatique (INRIA)January 2023, 17HAL
 42 miscAn equilibrated flux a posteriori error estimator for defeaturing problems.December 2023HAL
 43 miscAsymptotic optimality of the edge finite element approximation of the timeharmonic Maxwell's equations.September 2023HALback to text
 44 miscA stable local commuting projector and optimal hp approximation estimates in H(curl).December 2023HAL
 45 miscConstrained and unconstrained stable discrete minimizations for probust local reconstructions in vertex patches in the De Rham complex.May 2023HAL
 46 miscA hypocoercivityexploiting stabilised finite element method for Kolmogorov equation.January 2024HAL

47
misc
$hp$ optimal convergence of the original DG method for linear hyperbolic problems on special simplicial meshes.November 2023HALback to text  48 miscSpectral correctness of the discontinuous Galerkin approximation of the firstorder form of Maxwell's equations with discontinuous coefficients.June 2023HALback to text
 49 miscAsymptotic and invariantdomain preserving schemes for scalar conservation equations with stiff source terms and multiple equilibrium points.December 2023HALback to text
 50 miscConvergence analysis for the wave equation discretized with hybrid methods in space (HHO, HDG and WG) and the leapfrog scheme in time.September 2023HALback to text
 51 miscAdaptive regularization for the Richards equation.October 2023HAL
 52 miscRobust energy a posteriori estimates for nonlinear elliptic problems.March 2023HALback to text
 53 miscGuaranteed, locally efficient, and robust a posteriori estimates for nonlinear elliptic problems in iterationdependent norms. An orthogonal decomposition result based on iterative linearization.July 2023HAL
 54 miscA reduced basis method for frictional contact problems formulated with Nitsche's method.July 2023HALback to text
12.3 Cited publications
 55 inproceedingsScalable Domain Decomposition Preconditioners for Heterogeneous Elliptic Problems.Proceedings of the International Conference on High Performance Computing, Networking, Storage and AnalysisSC '13New York, NY, USADenver, ColoradoAssociation for Computing Machinery2013DOIback to text
 56 articleKSPHPDDM and PCHPDDM: Extending PETSc with advanced Krylov methods and robust multilevel overlapping Schwarz preconditioners.Computers & Mathematics with Applications842021, 277295DOIback to text