RESEARCH CENTRE

Inria Paris Centre

IN PARTNERSHIP WITH:

Ecole des Ponts ParisTech

2023
ACTIVITY REPORT

Project-Team

SERENA

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

Contents

Project-Team SERENA

1 Team members, visitors, external collaborators

2 Overall objectives

3 Research program

3.1 PDE level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Advanced numerical discretization methods . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Iterative linearization, domain decomposition, and multigrid solvers . . . . . . . . . . . . .
3.4 Reliability by a posteriori error control
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 Safe and correct programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 Application domains

4.1 Multiphase flows and transport of contaminants in the subsurface . . . . . . . . . . . . . . .
4.2 Industrial risks in energy production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 Social and environmental responsibility

5.1 Impact of research results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6 Highlights of the year

1

2

3

4
4
4
4
4
5

5
5
5

5
5

5

7 New software, platforms, open data

6
6
7.1 New software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
7.1.1 DiSk++ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
7.1.2 APS-MG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
7.1.3 FEMLAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
7.1.4 Skwer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
7.1.5 coq-num-analysis
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
7.1.6 MODFRAC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
7.1.7 nef-flow-fpm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
7.1.8 nef-transport-fpm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.1.9 ParaCirce . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
7.1.10 Pruners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
7.2 Open data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

8 New results

11
8.1 Research axis 1: Advanced numerical discretizations and solvers . . . . . . . . . . . . . . . . 11
8.2 Research axis 2: A posteriori error control, adaptivity, and safe and correct programming . 12
8.3 Research axis 3: Applications to environment and energy . . . . . . . . . . . . . . . . . . . . 14
8.4 Research axis 4: PDE and numerical analysis foundations . . . . . . . . . . . . . . . . . . . . 16

9 Bilateral contracts and grants with industry

19
9.1 Bilateral contracts with industry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

10 Partnerships and cooperations

10.1 International research visitors

19
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
10.1.1 Visits of international scientists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
10.1.2 Visits to international teams
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
10.3 National initiatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

10.2 European initiatives

10.2.1 H2020 projects

11 Dissemination

22
11.1 Promoting scientific activities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
11.1.1 Scientific events: organisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
11.1.2 Scientific events: selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
11.1.3 Journal
11.1.4 Invited talks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
11.1.5 Leadership within the scientific community . . . . . . . . . . . . . . . . . . . . . . . . 23
11.1.6 Scientific expertise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
11.1.7 Research administration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
11.2 Teaching - Supervision - Juries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
11.2.1 Teaching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
11.2.2 Supervision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
11.2.3 Juries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
11.3 Popularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
11.3.1 Internal or external Inria responsibilities . . . . . . . . . . . . . . . . . . . . . . . . . . 25
11.3.2 Interventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

12 Scientific production

25
12.1 Major publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
12.2 Publications of the year
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
12.3 Cited publications

Project SERENA

1

Project-Team SERENA

Creation of the Project-Team: 2017 April 01

Keywords

Computer sciences and digital sciences

A2.1.3. – Object-oriented 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

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Inria Annual Report 2023

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]

• Jean-Charles Gilbert [INRIA, Emeritus, from Oct 2023]

• Jean-Luc 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]

Post-Doctoral Fellows

• Akram Beni Hamad [INRIA, Post-Doctoral Fellow]

• Andre Harnist [INRIA, Post-Doctoral 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]

Project SERENA

3

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 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.

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Inria Annual Report 2023

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 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.

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 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.

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 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.

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 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.

Project SERENA

5

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
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.

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.

4 Application domains

4.1 Multiphase flows and transport of contaminants in the subsurface

• fractured and porous media

• flow in large-scale 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 CEA-EdF-Inria summer school on Robust Polyhedral Discretizations for Computational Mechanics
(June 26-30, 2023).

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Inria Annual Report 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 boundary-value
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 discon-
tinuous). Cell-based 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 high-
order Discontinuous Skeletal methods are Hybridizable Discontinuous Galerkin methods and the
recently-devised Hybrid High-Order methods. Some major benefits of Discontinuous Skeletal
methods are that their construction is dimension-independent and that they offer the possibility to
use general meshes with polytopal cells and non-matching 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 equa-
tions arising from the engineering sciences (mechanical, thermal, diffusion). The discretization is
based on the "Hybrid high-order" 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: https://github.com/wareHHOuse/diskpp

Publication: hal-01429292

Author: Matteo Cicuttin

Contact: Matteo Cicuttin

Partner: CERMICS

7.1.2 APS-MG

Name: A-Posteriori-Steered MultiGrid

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

P-robustness

Scientific Description: APS-MG (a-posteriori-steered multigrid) is a geometric-type multigrid solver
whose execution is steered by the associated a posteriori estimate of the algebraic error. In par-
ticular, the descent direction and the level-wise step sizes are adaptively optimized. APS-MG
corresponds to a V-cycle geometric multigrid with zero pre- and solely one post-smoothing step,
via block-Jacobi (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 discret-
ization, i.e., APS-MG contracts the error on each iteration by a factor that is independent of p.
APS-MG is the implementation of the solver developed in https://hal.science/hal-02070981 and
https://hal.science/hal-02494538.

Project SERENA

7

Functional Description: APS-MG (a-posteriori-steered multigrid) is an iterative linear solver implemen-
ted in MATLAB. It can treat systems of linear algebraic equations arising from order p conforming
finite element discretization of second-order elliptic diffusion problems. APS-MG is a geometric-
type multigrid method and uses a hierarchy of nested meshes. It corresponds to a V-cycle geometric
multigrid solver with zero pre- and one post-smoothing step via block-Jacobi (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: https://github.com/JanPapez/APS-MG

Publications: hal-02070981, hal-02494538, hal-02498247

Contact: Jan Papez

7.1.3 FEMLAB

Name: FEMLAB

Keywords: High order finite elements, Discontinuous Galerkin, Hybrid high-order 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: https://gitlab.inria.fr/zdong/FEMLAB

Publications: hal-03109470, hal-03109548, hal-03322267, hal-03513280, hal-03185683, hal-03315088,

hal-03695484, hal-03753221, hal-03889072

Contact: Zhaonan Dong

7.1.4 Skwer

Keywords: Differential equations, State-oriented simulation

Scientific Description: Unlike classical approaches which discretize time a priori to determine the state,
the State-Oriented 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 differential-algebraic equa-
tions (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 variable-equation 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 formula-based
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

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Inria Annual Report 2023

based on direct generation of exponential terms. We have tested the final algorithm over various
systems of differential equations to validate the approach.

URL: https://gitlab.inria.fr/skwer/skwer

Publication: hal-01636392

Contact: Sebastien Furic

Participants: Sebastien Furic, François Clement, Geraldine Pichot

7.1.5 coq-num-analysis

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 Lax-Milgram 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 long-term 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 rep-
resentation 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.

URL: https://lipn.univ-paris13.fr/coq-num-analysis/

Publications: hal-01344090, hal-01391578, hal-03105815, hal-03471095, hal-03516749, hal-03889276

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 Math-

ématiques 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 frontal-Delaunay approach

in a Riemannian context.

Project SERENA

9

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 poly-
gons 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 subseg-
ments, 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: hal-03480570, hal-02102811

Contact: Geraldine Pichot

Participants: Patrick Laug, Houman Borouchaki, Geraldine Pichot

Partner: Université de Technologie de Troyes

7.1.7 nef-flow-fpm

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 two-phase flow in frac-
tured 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 Por-
ous Media V. Martin, J. Jaffré, J. E. Roberts, SIAM Journal on Scientific Computing, 2005. ht-
tps://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.

Functional Description: nef-flow-fpm is a Matlab code to simulate flows in fractured porous media with

the mixed-hybrid 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: https://gitlab.inria.fr/nef/nef-flow-fpm

Contact: Geraldine Pichot

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

7.1.8 nef-transport-fpm

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

Scientific Description: The discretization in space is performed with a cell-centered finite volume

scheme. The discretization in time can be either explicit or implicit.

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Functional Description: nef-transport-fpm is a Matlab code for simulating transport by advection diffu-

sion in porous-fractured 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. Wa-
ter Resources Research, 29(8):2861-2869, 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 genera-

tion 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: https://gitlab.inria.fr/slegrand/paracirce

Publication: hal-03190252

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: https://team.inria.fr/serena/en/research/software/pruners/

Authors: Thierry Martinez, Simon Legrand, Geraldine Pichot

Contact: Thierry Martinez

Project SERENA

7.2 Open data

11

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.

8 New results

8.1 Research axis 1: Advanced numerical discretizations and solvers

Participants:

Zhaonan Dong, Alexandre Ern, Jean-Luc Guermond, Michel Kern,
Stefano Piccardo, Morgane Steins, Martin Vohralík, Zuodong Wang.

Invariant-domain time-stepping for compressible flows

Participants:

Alexandre Ern, Jean-Luc Guermond, Zuodong Wang.

Figure 1: Compressible Navier–Stokes simulation at Re = 1000 (from [26])

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 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 asymptotic-preserving.

Polytopal discretization methods

Participants:

Zhaonan Dong, Alexandre Ern, Jean-Luc Guermond, Stefano Piccardo,
Morgane Steins, Zuodong Wang.

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Inria Annual Report 2023

Further progress has been accomplished in the development and analysis of hybrid high-order (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 time-explicit 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.

Figure 2: A posteriori estimated error (left) and actual error (right) (from [20]). Nédélec finite element
approximation of the curl–curl (simplified Maxwell) problem, L-shaped 3D domain, top view (top) and
side view (bottom).

In [20], we present a local construction of H(curl)-conforming piecewise polynomials satisfying a
prescribed curl constraint. We start from a piecewise polynomial not contained in the H(curl) 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 best-approximations over the entire local versions of H(curl). This allows to design
guaranteed, fully computable, constant-free, and polynomial-degree-robust a posteriori error estimates of
the Prager–Synge type for Nédélec’s finite element approximations of the curl–curl problem. A divergence-
free decomposition of a divergence-free H(div)-conforming piecewise polynomial, relying on over-
constrained minimizations in Raviart–Thomas’ spaces, is the key ingredient. Numerical results confirm
the theoretical developments, see Figure 2 for an illustration.

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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., vector-valued mapped piecewise polynomials lying in the H(div)
space which appropriately approximate the desired divergence constraint. Our estimates are constant-
free 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 B-splines. 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.

Figure 3: Adaptive mesh refinement together with adaptive regularization for a nonsmooth nonlinearity
(from [28]).

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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 σ-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 coq-num-analysis (Section 7.1.5).

8.3 Research axis 3: Applications to environment and energy

Participants:

Alexandre Ern, Michel Kern, Simon Legrand, Clément Maradei, Aless-
andra 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 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 1.41 × 108 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

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Figure 4: Performances of the domain decomposition preconditioner HPDDM to solve flow in fractured
porous media

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 frac-
tures. 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.

Figure 5: Snapshot of the concentration in a discrete fracture network with 28 fractures.

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 two-phase flow. Six teams participated in the benchmark. The results

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Inria Annual Report 2023

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.

Figure 6: Evolution of the gas saturation (at 200 and 600 days). Gas is injected from the bottom left. The
middle section has a lower permeability, which forces the gas to rise above it.

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 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.

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 [18]. 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) [17].

8.4 Research axis 4: PDE and numerical analysis foundations

Participants:

Zhaonan Dong, Alexandre Ern, Jean-Luc Guermond, Géraldine Pichot,
Martin Vohralík.

A stable local commuting projector and optimal hp approximation estimates in H(curl)

Participants:

Théophile Chaumont-Frelet, Martin Vohralík.

Project SERENA

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Figure 7: Comparison of the wavefield for the diffusive damping (top) and the viscous damping (bottom)
at three different times, showing the expected phase reversal.

We design an operator from the infinite-dimensional Sobolev space H(curl) to its finite-dimensional
subspace formed by the Nédélec piecewise polynomials on a tetrahedral mesh that has the following
properties: 1) it is defined over the entire H(curl), 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 L2-norm, up to data oscillation; 5) it has optimal
(local-best) approximation properties; 6) it satisfies the commuting property with its sibling operator on
H(div); 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 H(curl) space. We in
particular employ it here to establish the two following results: i) equivalence of global-best, tangential-
trace- and curl-constrained, and local-best, unconstrained approximations in H(curl) including data
oscillation terms; and ii) fully h- and p- (mesh-size- and polynomial-degree-) optimal approximation
bounds valid under the minimal Sobolev regularity only requested elementwise.

Some novel results concerning Maxwell’s equations

Participants:

Alexandre Ern, Jean-Luc Guermond.

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].

Solutions to 1D advection-diffusion problems with discontinuous coefficients

Participants: Géraldine Pichot.

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Inria Annual Report 2023

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].

Model-order reduction

Participants:

Alexandre Ern, Abbas Kabalan.

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.

Best-approximation errors for fields with low regularity

Participants:

Zhaonan Dong, Alexandre Ern, Jean-Luc Guermond.

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.

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 first-order hyperbolic problems with constant convection fields in the L2 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.

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9 Bilateral contracts and grants with industry

9.1 Bilateral contracts with industry

Participants:

Alexandre Ern, Martin Vohralík.

• Two-part contract with CEA accompanying the PhD thesis of Nicolas Hugot.

• Two-part contract with Safran Tech accompanying the PhD thesis of Abbas Kabalan (co-supervised

with V. Ehrlacher).

• Two-part contract with CEA accompanying the PhD thesis of Romain Mottier.

• Two-part contract with ANDRA accompanying the PhD thesis of Ari Rappaport.

• Two-part 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. 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.

Other international visits to the team

Dirk Praetorius

Status researcher

Institution of origin: TU Wien

Country: Austria

Dates: june 2023

Context of the visit: Dirk Praetorius visited us in June 2023. He is a world-leading 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

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Fabio Vicini

Status researcher

Institution of origin: Politecnico di Torino

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 large-scale 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 non-matching
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

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10.2 European initiatives

10.2.1 H2020 projects

EMC2 EMC2 project on cordis.europa.eu

Title: Extreme-scale Mathematically-based 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 phys-
ics, 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.

10.3 National initiatives

Participants: Michel Kern.

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.

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Inria Annual Report 2023

11 Dissemination

Participants:

François Clément, Zhaonan Dong, Alexandre Ern, Gregor Gantner,
Jean-Luc 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 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.

11.1.2 Scientific events: selection

Member of the conference program committees Martin Vohralik was a member of the scientific com-
mittee 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 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 Technolo-

gies, Mühlheim an der Ruhr, Germany, August, 2023.

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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).

11.1.5 Leadership within the scientific community

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

• the Scientific Board of ORAP, Organisation Associative du Parallélisme;

• the board of École Doctorale Galilée at University Sorbonne Paris-Nord;

• 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 Interac-

tion 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 Paris-Nord,

France.

• Master: Martin Vohralík, Advanced finite elements, 21h, M1, ENSTA (Ecole nationale supérieure de

techniques avancées), Paris, France.

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Inria Annual Report 2023

11.2.2 Supervision

• PhD defended: Stefano Piccardo, Simulation of two-fluid 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 high-order 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 Novem-

ber 2023, Martin Vohralík.

• PhD in progress: Abbas Kabalan, Model order reduction for nonparametrized geometrical variabil-

ity, 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 high-order 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 approx-
imation of PDEs: regularization, linearization, discretization, and floating point precision, started
January 2021, Martin Vohralík.

• PhD in progress: Daniel Zegarra Vasquez, High-performance simulation of single-phase 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, April-September

2023, Martin Vohralík.

• Internship: Clément Maradei (Sup Galilée), Finite element simulation of the viscous wave equation,

February-March 2023, Michel Kern.

• Internship: Clément Maradei (Sup Galilée), Parallel adaptive hp-FEM, April-September 2023,

Zhaonan Dong and Martin Vohralík.

• Internship: Alessandra Marelli (Politecnico di Milano), Simulation of advective transport in fracture

networks, April-September 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 Franche-Comté), committee president for the PhD thesis of Georges
Seeman (Université Saint-Joseph de Beyrouth), and committee member for the PhD thesis of Sarah
Perez (Université de Pau et des Pays Adour) and Yipeng Wang (Sorbonne Université).

Project SERENA

25

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] E. Ahmed, S. Ali Hassan, C. Japhet, M. Kern and M. Vohralík. ‘A posteriori error estimates and
stopping criteria for space-time domain decomposition for two-phase flow between different
rock types’. In: SMAI Journal of Computational Mathematics 5 (Dec. 2019), pp. 195–227. DOI:
10.5802/smai-jcm.47. URL: https://hal.inria.fr/hal-01540956.

[2] L. Amir and M. Kern. ‘Preconditioning a coupled model for reactive transport in porous media’. In:
International Journal of Numerical Analysis and Modeling 16.1 (2019), pp. 18–48. URL: https://in
ria.hal.science/hal-01327307.

[3]

[4]

S. Boldo, F. Clément, F. Faissole, V. Martin and M. Mayero. ‘A Coq formal proof of the Lax–Milgram
theorem’. In: 6th ACM SIGPLAN Conference on Certified Programs and Proofs. Paris, France,
16th Jan. 2017. DOI: 10.1145/3018610.3018625. URL: https://hal.inria.fr/hal-01391578.

S. Boldo, F. Clément, J.-C. Filliâtre, M. Mayero, G. Melquiond and P. Weis. ‘Trusting computations:
A mechanized proof from partial differential equations to actual program’. In: Computers and
Mathematics with Applications 68.3 (Aug. 2014), pp. 325–352. DOI: 10.1016/j.camwa.2014.06.0
04. URL: http://dx.doi.org/10.1016/j.camwa.2014.06.004.

[5] E. Cancès, G. Dusson, Y. Maday, B. Stamm and M. Vohralík. ‘Guaranteed and robust a posteriori
bounds for Laplace eigenvalues and eigenvectors: conforming approximations’. In: SIAM Journal
on Numerical Analysis 55.5 (Sept. 2017), pp. 2228–2254. DOI: 10.1137/15M1038633. URL: https:
//hal.inria.fr/hal-01194364.

[6] A. Cangiani, Z. Dong and E. H. Georgoulis. ‘hp -Version discontinuous Galerkin methods on
essentially arbitrarily-shaped elements’. In: Mathematics of Computation 91.333 (Jan. 2022), pp. 1–
35. DOI: 10.1090/mcom/3667. URL: https://hal.inria.fr/hal-03109470.

[7] D. A. Di Pietro and A. Ern. ‘A hybrid high-order locking-free method for linear elasticity on general
meshes’. In: Comput. Methods Appl. Mech. Engrg. 283 (2015), pp. 1–21. DOI: 10.1016/j.cma.2014
.09.009. URL: http://dx.doi.org/10.1016/j.cma.2014.09.009.

[8] A. Ern, T. Gudi, I. Smears and M. Vohralík. ‘Equivalence of local-and global-best approximations, a
simple stable local commuting projector, and optimal hp approximation estimates in H (div)’. In:
IMA Journal of Numerical Analysis 42.2 (1st Apr. 2022), pp. 1023–1049. DOI: 10.1093/imanum/dra
a103. URL: https://hal.inria.fr/hal-02268960.

[9] A. Ern and J.-L. Guermond. ‘Finite element quasi-interpolation and best approximation’. In: ESAIM
Math. Model. Numer. Anal. 51.4 (2017), pp. 1367–1385. URL: https://doi.org/10.1051/m2an/2
016066.

[10] A. Ern, F. Hédin, G. Pichot and N. Pignet. Hybrid high-order methods for flow simulations in
extremely large discrete fracture networks. 18th Nov. 2022. URL: https://hal.inria.fr/hal-034
80570.

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Inria Annual Report 2023

[11] A. Ern and M. Vohralík. ‘Polynomial-degree-robust a posteriori estimates in a unified setting
for conforming, nonconforming, discontinuous Galerkin, and mixed discretizations’. In: SIAM
Journal on Numerical Analysis 53.2 (Apr. 2015), pp. 1058–1081. DOI: 10.1137/130950100. URL:
https://hal.inria.fr/hal-00921583.

[12] A. Ern and M. Vohralík. ‘Stable broken H1 and H(div) polynomial extensions for polynomial-
degree-robust potential and flux reconstruction in three space dimensions’. In: Mathematics of
Computation 89.322 (1st Mar. 2020), pp. 551–594. DOI: 10.1090/mcom/3482. URL: https://hal
.inria.fr/hal-01422204.

[13] T.-T.-P. Hoang, J. Jaffré, C. Japhet, M. Kern and J. E. Roberts. ‘Space-time domain decomposition
methods for diffusion problems in mixed formulations’. In: SIAM J. Numer. Anal. 51.6 (2013),
pp. 3532–3559. DOI: 10.1137/130914401. URL: http://dx.doi.org/10.1137/130914401.

12.2 Publications of the year

International journals

[14] E. Ahusborde, B. Amaziane, S. de Hoop, M. El Ossmani, E. Flauraud, F. P. Hamon, M. Kern, A.
Socié, D. Su, K. U. Mayer, M. Tóth and D. Voskov. ‘A benchmark study on reactive two-phase
flow in porous media: Part II -results and discussion’. In: Computational Geosciences (2023). URL:
https://hal.science/hal-04237832.

[15]

[16]

I. Ben Gharbia, J. Ferzly, M. Vohralík and S. Yousef. ‘Adaptive inexact smoothing Newton method
for a nonconforming discretization of a variational inequality’. In: Computers & Mathematics with
Applications 133 (2023), pp. 12–29. DOI: 10.1016/j.camwa.2022.11.031. URL: https://inria
.hal.science/hal-03696024.

I. Ben Gharbia, J. Ferzly, M. Vohralík and S. Yousef. ‘Semismooth and smoothing Newton methods
for nonlinear systems with complementarity constraints: Adaptivity and inexact resolution’. In:
Journal of Computational and Applied Mathematics 420 (2023), p. 114765. DOI: 10.1016/j.cam.2
022.114765. URL: https://inria.hal.science/hal-03355116.

[17] E. Burman, G. Delay and A. Ern. ‘The unique continuation problem for the heat equation discretized
with a high-order space-time nonconforming method’. In: SIAM Journal on Numerical Analysis
61.5 (2023), pp. 2534–2557. URL: https://hal.science/hal-03720960.

[18] E. Burman, G. Delay, A. Ern and L. Oksanen. ‘A stability estimate for data assimilation subject to
the heat equation with initial datum’. In: Comptes Rendus. Mathématique (2024). URL: https://h
al.science/hal-03830744.

[19] A. Cangiani, Z. Dong and E. H. Georgoulis. ‘A posteriori error estimates for discontinuous Galerkin
methods on polygonal and polyhedral meshes’. In: SIAM Journal on Numerical Analysis 61.5 (2023),
pp. 2352–2380. DOI: 10.1137/22M1516701. URL: https://inria.hal.science/hal-03753221
.

[20] T. Chaumont-Frelet and M. Vohralík. ‘p-robust equilibrated flux reconstruction in H(curl) based
on local minimizations. Application to a posteriori analysis of the curl-curl problem’. In: SIAM
Journal on Numerical Analysis 61.4 (17th July 2023), pp. 1783–1818. DOI: 10.1137/21M141909X.
URL: https://inria.hal.science/hal-03227570.

[21] P. Daniel and M. Vohralík. ‘Guaranteed contraction of adaptive inexact hp-refinement strategies
with realistic stopping criteria’. In: ESAIM: Mathematical Modelling and Numerical Analysis 57.1
(21st Feb. 2023), pp. 329–366. DOI: 10.1051/m2an/2022082. URL: https://inria.hal.science
/hal-02486433.

[22] Z. Dong and A. Ern. ‘C 0-hybrid high-order methods for biharmonic problems’. In: IMA Journal of
Numerical Analysis 44.1 (Feb. 2024), pp. 24–57. DOI: 10.1093/imanum/drad003. URL: https://i
nria.hal.science/hal-03695484.

[23] Z. Dong, A. Ern and J.-L. Guermond. ‘Local decay rates of best-approximation errors using vector-
valued finite elements for fields with low regularity and integrable curl or divergence’. In: Comptes
Rendus. Mathématique 361 (11th May 2023), pp. 723–736. DOI: 10 . 5802 / crmath . 347. URL:
https://inria.hal.science/hal-03513280.

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27

[24] Z. Dong, M. Hauck and R. Maier. ‘An improved high-order method for elliptic multiscale problems’.
In: SIAM Journal on Numerical Analysis 61.4 (2023), pp. 1918–1937. DOI: 10.1137/22M153392X.
URL: https://inria.hal.science/hal-03839664.

[25] Z. Dong and L. Mascotto. ‘hp-optimal interior penalty discontinuous Galerkin methods for the
biharmonic problem’. In: Journal of Scientific Computing 96.30 (18th May 2023). DOI: 10.48550/a
rXiv.2212.03735. URL: https://inria.hal.science/hal-03889072.

[26] A. Ern and J.-L. Guermond. ‘Invariant-domain preserving high-order time stepping: II. IMEX
schemes *’. In: SIAM Journal on Scientific Computing 45.5 (2023), A2511–A2538. URL: https://ha
l.science/hal-03703035.

[27] A. Ern and J.-L. Guermond. ‘The discontinuous Galerkin approximation of the grad-div and curl-
curl operators in first-order form is involution-preserving and spectrally correct’. In: SIAM Journal
on Numerical Analysis (2024). URL: https://hal.science/hal-04003475.

[28] F. Févotte, A. Rappaport and M. Vohralík. ‘Adaptive regularization, discretization, and linearization
for nonsmooth problems based on primal-dual gap estimators’. In: Computer Methods in Applied
Mechanics and Engineering 418 (24th May 2023), p. 116558. DOI: 10.1016/j.cma.2023.116558.
URL: https://hal.science/hal-04105560.

[29] G. Gantner and M. Vohralík. ‘Inexpensive polynomial-degree-robust equilibrated flux a posteriori
estimates for isogeometric analysis’. In: Mathematical Models and Methods in Applied Sciences
(2023), pp. 1–46. DOI: 10.1142/S0218202524500076. URL: https://inria.hal.science/hal-
03819048.

[30] S. de Hoop, D. Voskov, E. Ahusborde, B. Amaziane and M. Kern. ‘A benchmark study on reactive
two-phase flow in porous media: Part I -model description’. In: Computational Geosciences (2023).
URL: https://hal.science/hal-04237764.

[31] M. Jayadharan, M. Kern, M. Vohralík and I. Yotov. ‘A space-time multiscale mortar mixed finite
element method for parabolic equations’. In: SIAM Journal on Numerical Analysis 61.2 (3rd Apr.
2023), pp. 675–706. DOI: 10.1137/21M1447945. URL: https://inria.hal.science/hal-0335
5088.

[32] K. Mitra and M. Vohralík. ‘A posteriori error estimates for the Richards equation’. In: Mathematics

of Computation (Dec. 2023). URL: https://inria.hal.science/hal-03328944.

[33]

[34]

I. Niakh, G. Drouet, V. Ehrlacher and A. Ern. ‘Stable model reduction for linear variational inequalit-
ies with parameter-dependent constraints’. In: ESAIM: Mathematical Modelling and Numerical
Analysis 57.1 (2023), pp. 167–189. URL: https://hal.science/hal-03611982.

S. Piccardo and A. Ern. ‘Surface tension effects between two immiscible Stokes fluids: a computa-
tional study using unfitted hybrid high-order methods and a level-set scheme’. In: SMAI Journal of
Computational Mathematics 9 (2023), pp. 257–283. URL: https://hal.science/hal-04034898.

[35] M. Steins, A. Ern, O. Jamond and F. Drui. ‘Time-explicit Hybrid High-Order method for the nonlinear
acoustic wave equation’. In: ESAIM: Mathematical Modelling and Numerical Analysis 57.5 (2023),
pp. 2977–3006. URL: https://hal.science/hal-03960273.

International peer-reviewed conferences

[36] S. Boldo, F. Clément, V. Martin, M. Mayero and H. Mouhcine. ‘A Coq Formalization of Lebesgue
Induction Principle and Tonelli’s Theorem’. In: Proceedings of the 25th International Symposium on
Formal Methods. 25th International Symposium on Formal Methods (FM 2023). Vol. 14000. Lecture
Notes in Computer Science. Lübeck, Germany, 3rd Mar. 2023, pp. 39–55. DOI: 10.1007/978-3-03
1-27481-7_4. URL: https://inria.hal.science/hal-03889276.

Doctoral dissertations and habilitation theses

[37]

S. Piccardo. ‘Simulation of two-fluid immiscible Stokes flows using hybrid nonconforming methods
and geometrically unfitted meshes’. Ecole des Ponts ParisTech; Universitat Politècnica de Catalunya,
Barcelona (Spain), 4th Dec. 2023. URL: https://hal.science/tel-04425653.

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Inria Annual Report 2023

[38] M. Steins. ‘An explicit hybrid high-order method for structural dynamics’. ENPC - École des Ponts

ParisTech, 5th Dec. 2023. URL: https://hal.science/tel-04421265.

Reports & preprints

[39] E. Baioni, A. Lejay, G. Pichot and G. M. Porta. Modeling diffusion in discontinuous media under
generalized interface conditions: theory and algorithms. 19th July 2023. URL: https://inria.hal
.science/hal-04166559.

[40] E. Baioni, A. Lejay, G. Pichot and G. M. Porta. Random walk modeling of conductive heat transport

in discontinuous media. 18th July 2023. URL: https://inria.hal.science/hal-04166562.

[41] S. Boldo, F. Clément, V. Martin, M. Mayero and H. Mouhcine. Lebesgue Induction and Tonelli’s
Theorem in Coq. RR-9457. Institut National de Recherche en Informatique et en Automatique
(INRIA), 10th Jan. 2023, p. 17. URL: https://inria.hal.science/hal-03564379.

[42] A. Buffa, O. Chanon, D. Grappein, R. Vázquez and M. Vohralík. An equilibrated flux a posteriori
error estimator for defeaturing problems. 26th Dec. 2023. URL: https://inria.hal.science/ha
l-04379236.

[43] T. Chaumont-Frelet and A. Ern. Asymptotic optimality of the edge finite element approximation of
the time-harmonic Maxwell’s equations. 24th Sept. 2023. URL: https://inria.hal.science/ha
l-04216433.

[44] T. Chaumont-Frelet and M. Vohralík. A stable local commuting projector and optimal hp approxim-

ation estimates in H(curl). 5th Dec. 2023. URL: https://inria.hal.science/hal-03817302.

[45] T. Chaumont-Frelet and M. Vohralík. Constrained and unconstrained stable discrete minimizations
for p-robust local reconstructions in vertex patches in the De Rham complex. 11th May 2023. URL:
https://inria.hal.science/hal-03749682.

[46] Z. Dong, E. H. Georgoulis and P. J. Herbert. A hypocoercivity-exploiting stabilised finite element
method for Kolmogorov equation. 23rd Jan. 2024. URL: https://inria.hal.science/hal-0441
3333.

[47] Z. Dong and L. Mascotto. hp-optimal convergence of the original DG method for linear hyperbolic
problems on special simplicial meshes. 8th Nov. 2023. URL: https://inria.hal.science/hal-0
4252310.

[48] A. Ern and J.-L. Guermond. Spectral correctness of the discontinuous Galerkin approximation of
the first-order form of Maxwell’s equations with discontinuous coefficients. 29th June 2023. URL:
https://hal.science/hal-04145808.

[49] A. Ern, J.-L. Guermond and Z. Wang. Asymptotic and invariant-domain preserving schemes for
scalar conservation equations with stiff source terms and multiple equilibrium points. 21st Dec.
2023. URL: https://hal.science/hal-04357751.

[50] A. Ern and M. Steins. Convergence analysis for the wave equation discretized with hybrid methods in
space (HHO, HDG and WG) and the leapfrog scheme in time. 8th Sept. 2023. URL: https://hal.sc
ience/hal-04200966.

[51] F. Févotte, A. Rappaport and M. Vohralík. Adaptive regularization for the Richards equation. 31st Oct.

2023. URL: https://hal.science/hal-04266827.

[52] A. Harnist, K. Mitra, A. Rappaport and M. Vohralík. Robust energy a posteriori estimates for nonlinear

elliptic problems. 17th Mar. 2023. URL: https://hal.science/hal-04033438.

[53] K. Mitra and M. Vohralík. Guaranteed, locally efficient, and robust a posteriori estimates for nonlin-
ear elliptic problems in iteration-dependent norms. An orthogonal decomposition result based on
iterative linearization. 9th July 2023. URL: https://inria.hal.science/hal-04156711.

[54]

I. Niakh, G. Drouet, V. Ehrlacher and A. Ern. A reduced basis method for frictional contact problems
formulated with Nitsche’s method. 21st July 2023. URL: https://inria.hal.science/hal-0416
8418.

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12.3 Cited publications

[55] P. Jolivet, F. Hecht, F. Nataf and C. Prud’homme. ‘Scalable Domain Decomposition Preconditioners
for Heterogeneous Elliptic Problems’. In: Proceedings of the International Conference on High
Performance Computing, Networking, Storage and Analysis. SC ’13. Denver, Colorado: Association
for Computing Machinery, 2013. DOI: 10.1145/2503210.2503212.

[56] P. Jolivet, J. E. Roman and S. Zampini. ‘KSPHPDDM and PCHPDDM: Extending PETSc with ad-
vanced Krylov methods and robust multilevel overlapping Schwarz preconditioners’. In: Computers
& Mathematics with Applications 84 (2021), pp. 277–295. DOI: https://doi.org/10.1016/j.ca
mwa.2021.01.003.
<scratchpad>
- Main topic: Numerical methods for environmental problems
- Key points: Development of numerical algorithms for simulation of multiphase flows, transport of contaminants, and industrial risks in energy production
- Structure: The document is organized into several sections, including activity report, project-team, research program, application domains, social and environmental responsibility, new software, and partnerships and cooperations
- Potential artifacts to ignore: Links, HTML tags, and other web-related elements
</scratchpad>

After analyzing the document, I have identified the following key points:

* The SERENA project-team is concerned with numerical methods for environmental problems, specifically simulation of multiphase flows, transport of contaminants, and industrial risks in energy production.
* The team has developed several numerical algorithms, including DiSk++, APS-MG, FEMLAB, Skwer, coq-num-analysis, MODFRAC, nef-flow-fpm, nef-transport-fpm, ParaCirce, and Pruners.
* The team has also worked on various research axes, including advanced numerical discretizations and solvers, a posteriori error control, adaptivity, and safe and correct programming.
* The team has collaborated with several industry partners, including EDF, CEA, IFP Energies Nouvelles, ANDRA, ITASCA, and BRGM.
* The team has also participated in several international research initiatives, including the H2020 project EMC2 and the European initiative on Domain Decomposition Preconditioners.

<final_summary>
The SERENA project-team is a research group focused on developing numerical methods for environmental problems, such as simulation of multiphase flows, transport of contaminants, and industrial risks in energy production. The team has developed several numerical algorithms, including DiSk++, APS-MG, FEMLAB, and coq-num-analysis, and has worked on various research axes, including advanced numerical discretizations and solvers, a posteriori error control, adaptivity, and safe and correct programming. The team has collaborated with industry partners and participated in international research initiatives, demonstrating its commitment to addressing environmental challenges through numerical simulation.
The SERENA project-team is a research group focused on developing numerical methods for environmental problems, such as simulation of multiphase flows, transport of contaminants, and industrial risks in energy production. The team has developed several numerical algorithms, including DiSk++, APS-MG, FEMLAB, and coq-num-analysis, and has worked on various research axes, including advanced numerical discretizations and solvers, a posteriori error control, adaptivity, and safe and correct programming. The team has collaborated with industry partners and participated in international research initiatives, demonstrating its commitment to addressing environmental challenges through numerical simulation.