SERENA

SERENA - 2025

2025Activity report‌Project-TeamSERENA

RNSR: 201521772E‌‌

Research center Inria Paris Centre
In partnership with:‌Ecole Nationale des Ponts‌ et Chaussées
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)

Creation‌ of the Project-Team: 2017‌ April 01

Each year,‌‌ Inria research teams publish an Activity Report presenting‌ their work and results‌ over the reporting period.‌‌ These reports follow a‌ common structure, with some optional sections depending on‌ the specific team. They typically begin by outlining‌ the overall objectives and research programme, including the‌ main research themes, goals, and methodological approaches. They‌ also describe the application domains targeted by the‌ team, highlighting the scientific or societal contexts in‌ which their work is situated.

The reports then‌ present the highlights of the year, covering major‌ scientific achievements, software developments, or teaching contributions. When‌ relevant, they include sections on software, platforms, and‌ open data, detailing the tools developed and how‌ they are shared. A substantial part is dedicated‌ to new results, where scientific contributions are described‌ in detail, often with subsections specifying participants and‌ associated keywords.

Finally, the Activity Report addresses funding,‌ contracts, partnerships, and collaborations at various levels, from‌ industrial agreements to international cooperations. It also covers‌ dissemination and teaching activities, such as participation in‌ scientific events, outreach, and supervision. The document concludes‌ with a presentation of scientific production, including major‌ publications and those produced during the year.

Keywords‌

Computer Science and Digital Science

A2.1.3. Object-oriented programming‌
A2.1.4. Functional programming
A4.5.3. Program proof
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

1 Team members, visitors, external collaborators

Research Scientists‌

Martin Vohralík [Team leader, INRIA,‌ Senior Researcher, HDR]
François Clement [‌INRIA, Researcher]
Zhaonan Dong [INRIA‌, Researcher]
Alexandre Ern [Ecole des‌ Ponts (ParisTech), Senior Researcher, HDR]‌
Jean-Charles Gilbert [INRIA, Emeritus]
Michel‌ Kern [INRIA, Researcher]
Géraldine Pichot‌ [INRIA, Researcher]

Post-Doctoral Fellows

Divay‌ Garg [INRIA, Post-Doctoral Fellow, until‌ May 2025]
Rekha Khot [INRIA,‌ Post-Doctoral Fellow, until Nov 2025]
Ibtissem‌ Lannabi [INRIA, Post-Doctoral Fellow]
Lukas‌ Renelt [INRIA, Post-Doctoral Fellow, from‌ Jun 2025]
Tanvi Wadhawan [INRIA,‌ Post-Doctoral Fellow, from Sep 2025]

PhD‌ Students

Nicolas Hugot [CEA]
Abbas Kabalan‌ [SafranTech, until Oct 2025]
Clement‌ Maradei [INRIA]
Romain Mottier [ENPC‌, until Jul 2025]
Baptiste Plaquevent-Jourdain [‌INRIA, until Sep 2025]
Bahaa Eddine‌ Sidi Hida [EDF]
Zuodong Wang [‌INRIA, until Sep 2025]
Daniel Zegarra‌ Vasquez [INRIA, until May 2025]‌
Benjamin Zurich [INRIA, from Jun 2025‌]

Technical Staff

Simon Legrand [INRIA]‌
Raphael Zanella [INRIA, Engineer, from‌ Oct 2025]

Administrative Assistant

Derya Gok [‌INRIA]

Visiting Scientists

Bernardo Cockburn [University of Minnesota, from‌ Nov 2025 until Nov‌ 2025]
Jean-Pierre Dussault‌‌ [UNIV SHERBROOKE, from Jun 2025 until‌ Sep 2025]
Jean-Luc‌ Guermond [Texas A&M‌‌ University, from May 2025 until Jul 2025‌]
Buyang Li [‌Hong Kong Polytechnic University‌‌, from Jun 2025 until Jul 2025]‌
Peter Moritz Von Schultzendorff‌ [Univ Bergen,‌‌ until Feb 2025]

External Collaborators

Guy Chavent‌ [retired from Inria‌]
François Delebecque [‌‌retired from Inria]
Sebastien Furic [indépendant‌, until Oct 2025‌]
Jérôme Jaffré [‌‌retired from Inria, HDR]
Habib Jreige‌ [SciWorks]
Vincent‌ Martin [UTC]‌‌
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.

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 Optimization,‌ complementarity, nonsmooth analysis, discrete geometry

We dedicate a‌ specific effort to solvers of algebraic inequalities in‌ complementarity form. A rather general mathematical formulation of‌ a complementarity problem reads: find a point $x‌ \in ℝ^{n}$ such that $F (x‌) ⩾ 0$ , $G (x)‌ ⩾ 0$ , and $F_{i} (x‌) G_{i} ( x) = 0‌$ for all $i \in {1, ...‌, n}$ , or, equivalently,

0 ⩽​‌﻿﻿ F (x)​​﻿﻿ ⊥ G (x​​​‌) ⩾ 0 .﻿​﻿﻿

Here, $F$ and $G‌ : ℝ^{n} \mapsto ℝ^{n}$ are smooth‌ functions, the inequalities are understood componentwise, and “‌ $⊥$ ” denotes perpendicularity with respect to the‌ Euclidean scalar product. The orthogonality conditions highlight the‌ combinatorial aspect of the problem: there are ${2‌}^{n}$ ways of realizing them. Determining whether a‌ linear complementarity problem (i.e., $F$ and $G$ are‌ affine) with integer data has a rational solution,‌ let alone calculating one, is NP complete.

Complementarity‌ is used to mathematically formulate systems in which‌ one model among several is active at a‌ given place and time (the index $i$ above);‌ hence, the active model ( $F_{i} (‌ x) = 0$ or $G_{i} (‌ x) = 0$ ) is an unknown‌ of the problem. They generalize the first-order optimality‌ conditions of an optimization problem, in which complementarity‌ occurs between the inequality constraints and their optimal‌ multipliers. Examples of applications are found in nonsmooth‌ mechanics and dynamics, to model the phase transition‌ problem in multiphase flows, precipitation-dissolution problems in chemistry,‌ portfolio management in finance, computer graphics, meteorology, or‌ economic equilibrium to mention a few. These problems‌ can be reformulated as (usually) nonsmooth equations, which‌ partly motivates our interest in nonsmooth analysis.

3.5‌ Reliability by a posteriori error control

The fifth‌ 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.

3.6‌‌ 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.‌‌

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

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, mainly‌ 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

We have achieved‌‌ in 60 a first construction of projectors on‌ the canonical finite element‌ subspaces of the ${H‌‌}^{1}$ , $H ( curl)$ , and‌ $H (div)‌$ Sobolev spaces that are‌‌ $L^{2}$ stable, commute with the gradient, curl,‌ and divergence differential operators,‌ and are locally defined‌‌ (A. Ern and M.‌ Vohralík with the colleagues J. Guzmán and P.‌ Potu from the Brown University).

The book chapter‌ 37 makes an introduction to the theory and‌ practice of a posteriori error estimates, allowing to‌ give bounds on the error between the unknown‌ solution of a partial differential equation and its‌ numerical approximation (M. Vohralík with the colleague S.‌ Yousef from IFP Energies Nouvelles, 130 pages).‌

7 Latest software developments, platforms, open data

7.1‌ Latest software developments

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 discontinuous).‌ 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 equations 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
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 particular, 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 discretization,‌ 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.
Functional Description:
APS-MG (a-posteriori-steered‌ 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 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 rocq-num-analysis

Name:
Numerical analysis‌ Rocq library
Keywords:
Formal‌‌ methods, Rocq, Numerical analysis, Finite element modelling
Scientific‌ Description:
These Coq/Rocq developments‌ are based on the‌‌ Coquelicot library for real analysis, and on parts‌ of the Mathematical Components‌ library. They are based‌‌ on classical logic. Version 2.1 includes the formalization‌ and proof of: (1)‌ results about subsets, functions,‌‌ homogeneous binary relations, and finite families, (2) results‌ in algebra about commutative‌ monoids and groups, rings,‌‌ vector and affine spaces, including functions to a‌ given algrebraic structure, iterated‌ operations (sum, linear combination,‌‌ and barycenter), morphisms, algebraic substructures, and the specific‌ case of finite dimension‌ with the dimension and‌‌ the incomplete basis theorems, (3) the Lax-Milgram theorem,‌ including results from linear‌ algebra, geometry, functional analysis‌‌ and Hilbert spaces, (4) 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,‌‌ (5) simplicial Lagrange finite elements in any dimension‌ and of any degre‌ of approximation.
Functional Description:‌‌
Formal developments and proofs in Rocq 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:‌‌
Several parts of the‌ formalization in Coq/Rocq of simplicial Lagrange finite elements‌ have been generalized, and the building of the‌ FE is now almost exclusively based on affine‌ properties involving barycenters. Opam packages (v2.0, 17/06/2025) are‌ provided for each part of the library: subset,‌ algebra, lebesgue, lax-milgram, and fem. A paper about‌ the formalization of simplicial Lagrange finite elements is‌ submitted. The version 2.1 of the Opam packages‌ (24/11/2025) provides many modifications and new results about‌ binary relations and orders. A paper about the‌ formalization of monomial and graded orders is submitted.‌
URL:
https://lipn.univ-paris13.fr/rocq-num-analysis/
Publications:
hal-01344090, hal-01391578, hal-03105815‌, hal-03471095, hal-03516749, hal-03889276, hal-04713897‌, tel-04884651, hal-05128954, hal-05396788
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), UTC

7.1.5 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.
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:
Implementation of‌ a new automatic correction procedure for highly intricate‌ configurations, with the aim of keeping points sufficiently‌ separated to enable the volume mesh generation.
Publications:‌
hal-03480570, hal-02102811, hal-05029652v1, hal-05029676v2
Contact:‌
Geraldine Pichot
Participants:
Patrick Laug, Houman Borouchaki, Geraldine‌ Pichot
Partner:
Université de Technologie de Troyes

7.1.6‌ 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 fractured media‌ Hussein Hoteit, Abbas Firoozabadi, Advances in Water Resources‌ 31, 891–905, 2008. https://doi.org/10.1016/j.advwatres.2008.02.004

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

Validation benchmark test from‌ the publication: Inga Berre, et al., Verification benchmarks‌ for single-phase flow in three-dimensional fractured porous media,‌ Advances in Water Resources, Volume 147, 2021. https://doi.org/10.1016/j.advwatres.2020.103759.‌
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:
Validation of‌ the code by comparison‌ with an analytical solution‌‌ described in Brenner, K., Groza, M., Guichard, C.‌ et al. Gradient discretization‌ of hybrid dimensional Darcy‌‌ flows in fractured porous media. Numer. Math. 134,‌ 569–609 (2016). https://doi.org/10.1007/s00211-015-0782-x Add‌ new validation test cases‌‌ from Berre, et al., 2021. https://doi.org/10.1016/j.advwatres.2020.103759.
URL:
https://gitlab.inria.fr/nef/nef-flow-fpm‌
Publications:
hal-05029638v2, hal-05029676v2‌, hal-05029652v1
Contact:
Geraldine‌‌ Pichot
Participants:
Geraldine Pichot, Daniel Zegarra Vasquez, Michel‌ Kern, Raphael Zanella

7.1.7‌ demonstrator-nef-hpddm

Keywords:
Fracture network,‌‌ Geophysical flows, Linear Systems Solver
Functional Description:

The‌ demonstrator is a C++‌ code based on the‌‌ PETSc (Portable, Extensible Toolkit for Scientific Computation) library‌ which is able to‌ solve the linear system‌‌ A x = b, where A is an‌ invertible symmetric matrix, x‌ is the unknown vector‌‌ and b is a given vector. The goal‌ is to solve the‌ linear system assembled by‌‌ the Matlab finite element code nef-flow-fpm, which is‌ dedicated to the simulation‌ of flows in fractured/porous‌‌ media, with more performant solvers than those available‌ in Matlab.

The linear‌ system in the demonstrator‌‌ is loaded from binary files (.dat) or assembled‌ from structures loaded from‌ binary files. The binary‌‌ files are written in nef-flow-fpm with a specific‌ Matlab function available in‌ the PETSc repository (PetscBinaryWrite.m).‌‌ The solution obtained with the demonstrator may be‌ written in a binary‌ file through a PETSc‌‌ option (-ksp_view_solution), to be latter on loaded in‌ nef-flow-fpm for post-processing.

The‌ solving may be performed‌‌ in parallel on distributed memory machines thanks to‌ PETSc, which uses MPI‌ functions.

Any linear solver‌‌ and preconditioner available in PETSc may be chosen‌ to solve the linear‌ system. In particular, it‌‌ is possible to use the preconditioner KSPHPDDM for‌ domain decomposition methods. It‌ is then possible to‌‌ use Neumann matrices to improve the solving. A‌ Neumann matrix is the‌ matrix obtained for a‌‌ given subdomain (with overlap) where Neumann boundary conditions‌ are enforced at the‌ interfaces with the other‌‌ subdomains.

The source files of the demonstrator are‌ main.cpp, functions.hpp/cpp and scaling.hpp/cpp.‌
News of the Year:‌‌

[2025] Improvement of input files reading: a unique‌ input file per type‌ is now required. The‌‌ development of a NumPEx mini-app has started: https://github.com/numpex/apps-petsc‌

[2024] Development of the‌ demonstrator. Compilation through Make‌‌ or CMake. Non-regression tests, showing the good behavior‌ of the code to‌ solve flow in fracture‌‌ networks (2D), in porous media (3D) and fractured‌ porous media (2D/3D). CI‌ testing enabled through the‌‌ .gitlab-ci.yml. Readme and Latex documentation.
URL:
https://gitlab.inria.fr/nef/demonstrator-nef-hpddm
Contact:‌
Geraldine Pichot
Participants:
Geraldine‌ Pichot, Raphael Zanella

7.1.8‌‌ 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):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 generation‌ in parallel of 2D or 3D Gaussian Random‌ Fields (second order stationary).
Release Contributions:
- Singleton‌ handles MPI and FFTW global states - Data‌ distribution of the GRF is multidimensional - Better‌ logging facilities - .deb package generation
News of‌ the Year:
- MPI and FFTW global states‌ handled by a singleton - Data distribution of‌ the GRF is multidimensional - Logging facilities -‌ .deb package generation
URL:
https://gitlab.inria.fr/slegrand/paracirce
Publication:
hal-03190252
Contact:‌
Geraldine Pichot
Participants:
Geraldine Pichot, Simon Legrand

7.1.9‌ Hnm4lcp

Name:
Hybrid Newton-Min algorithm for Linear Complementarity‌ Problems
Keywords:
Linear complementarity problem, Global convergence, Least-square‌ merit function, Linesearch, Minimum function, Nonsmooth reformulation, P-matrix,‌ Polyhedral Newton-min algorithm, Semismooth Newton
Functional Description:

Hnm4lcp‌ solves a linear complementarity problem, which consists in‌ finding a vector x with nonnegative components such‌ that the vector Mx+q (the square matrix M‌ and the vector q are given) is also‌ nonnegative and perpendicular to x. This is denoted‌ by <blockquote> 0 <= x _|_ (Mx+q) >=‌ 0. </blockquote> This type of problem appears in‌ the optimality conditions of a quadratic optimization problem,‌ in nonsmooth mechanics (in robotics in particular), can‌ express dissolution-precipitation phenomena in chemistry, in multiphase flows,‌ in meteorology, etc.

The followed solution approach consists‌ in reformulating the problem in the form of‌ the nonsmooth equation H(x) = 0, where <blockquote>‌ H(x) := min(x,Mx+q) </blockquote> and to apply a‌ kind of semismooth Newton algorithm to it (the‌ used Jacobian belongs to the product B-differential of‌ H). This one is modified to ensure the‌ global convergence of the algorithm, which requires to‌ compute a point in a convex polyhedron at‌ some iterations (rarely, actually).
URL:
https://who.rocq.inria.fr/Jean-Charles.Gilbert/codes/hnm4lcp/hnm4lcp.html
Publication:
hal-02306526‌
Contact:
Jean Charles Gilbert
Partner:
Jean-Pierre Dussault, Université‌ de Sherbrooke, Québec, Canada

7.1.10 ISF

Name:
Isf‌ and Bdiffmin
Keywords:
B-differential, Hyperplane arrangement
Scientific Description:‌
The code is described in the following two‌ reports:- - https://inria.hal.science/hal-04048393 - https://hal.science/hal-04102933
Functional Description:

The‌ B-diffrential computed by Bdiffmin is a concept of‌ derivative for a nonsmooth function. The function considered‌ by Bdiffmin is the componentwise minimum of two‌ affine functions: x -> min (Ax+a,Bx+b), where A‌ and B are mxn real matrices, while a‌ and b are real vectors of size m.‌ In this case, the B-differential is a finite‌ (between 1 and 2m̂ éléments) collection of Jacobians‌ (i.e., mxn matrices). The computing time is polynomial‌ per computed Jacobian.

To realize this task, Bdiffmin‌ calls the Matlab function ISF (for Incremental Sign‌ Feasibility), which has been designed to determine the‌ chambers of an arrangement of hyperplanes having a‌ point in common. The sign vectors computed by‌ the latter function can be used to solve‌ a number of other enumeration problems such as - determining the signed‌ feasibility of strict inequality‌ systems, - listing the‌‌ orthants encountered by the null space of a‌ matrix, - itemizing the‌ pointed cones generated by‌‌ a set of vectors and their inverses, -‌ giving the bipartitions of‌ a finite set of‌‌ points that can be separated by an affine‌ hyperplane and many others.‌
URL:
https://who.rocq.inria.fr/Jean-Charles.Gilbert/codes/isf-bdiffmin/isf-bdiffmin.html
Publications:
hal-04048393‌‌, hal-04102933
Contact:
Jean Charles Gilbert
Participants:
Jean-Pierre‌ Dussault, Jean Charles Gilbert,‌ Baptiste Plaquevent-Jourdain

7.1.11 ISF_jl‌‌

Name:
Incremental Sign Feasibility - Julia version
Keywords:‌
Hyperplane arrangement, Combinatorics, Matroids‌
Functional Description:
This code‌‌ extends the ISF code (in Matlab) to general‌ affine arrangements, while ISF‌ is restricted to linear‌‌ arrangements. It is written in Julia. Its goal‌ is to list the‌ chambers of a hyperplane‌‌ arrangement, i.e., to obtain the sign vectors corresponding‌ to each of them.‌ This tool, mainly developed‌‌ for research purposes, may use several methods. In‌ particular, a "dual" approach‌ using the matroid circuits‌‌ associated with the arrangement is available.
Contact:
Baptiste‌ Plaquevent-Jourdain

8 New results‌

8.1 Research axis 1:‌‌ Advanced numerical discretizations and solvers

Participants: Zhaonan Dong‌, Jean-Pierre Dussault,‌ Alexandre Ern, Jean‌‌ Charles Gilbert, Jean-Luc Guermond, Rekha Khot‌, Clément Maradei,‌ Baptiste Plaquevent-Jourdain, Martin‌‌ Vohralík, Zuodong Wang.

Development and analysis‌ of HHO methods

Participants:‌ Zhaonan Dong, Alexandre‌‌ Ern, Rekha Khot, Romain Mottier.‌

Hybrid high-order (HHO) methods‌ play still an important‌‌ role in our research endeavors. In 47,‌ we have shown improved‌ $L^{2}$ -error estimates‌‌ on the time-integrated primal variable for the wave‌ equation in first-order form.‌ The analysis turned out‌‌ to be rather challenging and required us to‌ derive a novel interpolation‌ operator of broad interest‌‌ that can be applied to many hybrid nonconforming‌ methods, including HDG and‌ weak Galerkin methods. In‌‌ 44, we devised a novel unfitted HHO‌ method that can be‌ used to approximate eliptic‌‌ interface problems using unfitted meshes. The novel idea‌ is to stabilize poorly‌ cut cells by polynomial‌‌ extensions. We derived optimal error estimates and showed‌ numerically that the method‌ is well behaved. An‌‌ illustration of the problem setting and the outcome‌ of the pairing procedure‌ for stabilization by polynomial‌‌ extension is presented in Figure 1.

Figure 1.a — Figure 1:‌ Left: flower-like interface and‌ unfitted mesh. Right: outcome‌‌ of pairing procedure for stabilization by polynomial extension.‌

Figure 1.b — Figure 1:‌ Left: flower-like interface and‌ unfitted mesh. Right: outcome‌‌ of pairing procedure for stabilization by polynomial extension.‌

In 32, we‌ devised an HHO method‌‌ to approximate in space elasto-acoustic wave problems in‌ first-order form. The time‌ discretization is handled by‌‌ implicit or explicit Runge–Kutta methods. Interestingly, we showed‌ that implicit methods call‌ for a local elimination‌‌ of the cell unknowns, whereas explicit methods call‌ for the local elimination‌ of the face unknowns.‌‌ We notice that this latter elimination can still‌ be performed locally provided‌ a mixed-order polynomial setting‌‌ is employed so that‌ the stabilization operator is block-diagonal. An illustration is‌ presented in Figure 2, showcasing the propagation‌ of a Ricker wavelet in a two-layered medium‌ composed of granite and water.

Figure 2: Ricker‌ wavelet computed using the HHO method.

Time-stepping methods‌

Participants: Alexandre Ern, Jean-Luc Guermond.

In‌ 29, we devised third-order sectorially A-stable alternating‌ implicit Runge–Kutta (IRK) schemes. One traditional way to‌ split two stiff operators consists of using methods‌ like the Peaceman–Rachford alternating direction implicit method (ADI)‌ at the time-discrete level. This method combines two‌ two-stage, second-order IRK methods, one for each stiff‌ operator, in such a way that, at every‌ stage, only one of the two operators is‌ treated implicitly. The present contribution provides the first‌ generalization of this approach to third-order. We proved‌ that it is not possible to achieve third-order‌ accuracy by combining two four-stage IRK schemes, and‌ showed that this can be achieved by combining‌ two six-stage IRK schemes and we explicitly constructed‌ two examples. Moreover, we established sectorial A-stability for‌ each example. In addition, for each example, we‌ devised a companion explicit Runge–Kutta scheme that can‌ be used as a companion time-stepping scheme to‌ handle a third nonstiff operator. Finally, we showed‌ by numerical examples including two-dimensional nonlinear transport problems‌ discretized in space using finite elements that the‌ proposed schemes behave well.

In 59, we‌ showed that the well-established SUPG paradigm can be‌ successfully combined with an explicit Runge–Kutta (ERK) method.‌ SUPG is a popular approach to stabilize the‌ finite element approximation of steady linear transport equations.‌ Its application to time-dependent problems is, however, not‌ so straightforward, and, so far, only positive results‌ were available in the literature when using low-order‌ time-stepping schemes, e.g., the forward Euler method. Our‌ work provides a step forward by establishing stability‌ when SUPG is combined with third- and fourth-order‌ ERK methods.

Invariant-domain preserving discretizations

Participants: Zhaonan Dong‌, Alexandre Ern, Jean-Luc Guermond, Zuodong‌ Wang.

In 24, we devised a‌ new bound-preserving finite element scheme combined with an‌ implicit Euler time discretization to approximate the Allen–Cahn‌ equation. This equation constitutes a well-established approach to‌ study phase-field models. In our work, we derived‌ error estimates for our schemes with only polynomial‌ dependence on the small stiffness parameter characterizing the‌ width of the internal layers. This is the‌ first time that a bound-preserving method is analyzed‌ with such sharp error bounds.

In 61,‌ we devised a new mass conservative limiting and‌ showed how it can be applied to the‌ approximation of the steady-state radiation transport equations. The‌ main advantage of the new limiter is to‌ avoid the use of two simultaneous approximations as‌ in the widely used FCT (Flux Corrected Transport)‌ paradigm.

Complementarity problems

Participants: Jean-Pierre Dussault, Jean‌ Charles Gilbert, Baptiste Plaquevent-Jourdain.

The contribution‌ 26 proposes new approaches to solve the complementarity problem with some global‌ convergence result (with the‌ Polyhedral Newton-Min or PNM‌‌ algorithm) and fast local convergence (with a hybrid‌ version, called HNM). The‌ proposed approach reformulates the‌‌ problem with the Minimum C-function, that is $H‌ (x) :‌ = min (F‌‌ (x), G (x)‌) = 0$ ,‌ and takes the associated‌‌ least-square function $x \mapsto θ (x)‌ : = \frac{1}{2‌} {∥ H (x‌‌) ∥}_{2}^{2}$ as merit function, which‌ means that the progress‌ to the solution is‌‌ measured by the imposed decrease of $θ$ .‌ This approach is standard‌ when $H$ is smooth,‌‌ which is not the case here. In the‌ present case, this approach‌ is still very efficient‌‌ when it works... but many pitfalls must be‌ overcome to get that‌ efficiency. The main difficulty‌‌ is that the semi-smooth-like Newton direction on $H‌$ may not be a‌ descent direction of $θ‌‌$ . To overcome this dificulty, the semi-smooth-like Newton‌ direction is replaced at‌ some iterations by the‌‌ computation of a direction in a polyhedron defined‌ by a usually small‌ number of linear inequality‌‌ constraints (PNM principle). The HNM method has quadratic‌ local convergence when the‌ limit point satisfies some‌‌ reinforced regularity conditions. Numerical experiments with Hnm4lcp (§‌ 7.1.9) show the‌ efficiency of the approach‌‌ for linear complementarity problems (i.e., when $F$ and‌ $G$ are affine).

The‌ Levenberg-Marquardt approach is a‌‌ technique that makes possible to have global convergence‌ of Newton-like iterations with‌ much weaker regularity conditions,‌‌ hence making the convergence more robust. In 39‌, this approach is‌ applied to the nonsmooth‌‌ system $H (x) = 0$ for‌ the function $H$ defined‌ above. With this technique,‌‌ the PNM principe no longer needs to require‌ that the constructed polyhedra‌ are nonempty. Instead, the‌‌ algorithm minimizes a function made of convex quadratic‌ pieces, which always has‌ a minimizer, at the‌‌ price of a higher computational cost. Some complexity‌ issues are also evoked,‌ in relation with geometric‌‌ considerations close to hyperplane arrangements, which shed some‌ new light on the‌ reformulation of complementarity problems‌‌ with the minimum C-function. It is shown that‌ detecting strong stationarity of‌ the merit function $θ‌‌$ at a point (i.e., $θ^{'} (x‌; d) ⩾‌ 0$ , for all‌‌ $d$ ) is co-NP-complete. Finding a point satisfying‌ a weaker form of‌ stationarity (i.e., $0 \in‌‌ \partial_{C} θ ( x)$ ), however,‌ may be feasible under‌ suitable assumptions.

Discrete geometry‌‌

Participants: Jean-Pierre Dussault, Jean Charles Gilbert,‌ Baptiste Plaquevent-Jourdain.

The‌ hyperplanes in 27 are‌‌ linear, meaning that they all contain zero.‌ The contribution 57 extends‌ this approach proposed to‌‌ affine arrangements, which means that their hyperplanes may‌ not have a point‌ in common. Many properties‌‌ of affine arrangements are also stated in this‌ more general case and‌ proved with an analytic‌‌ viewpoint, whilst algebraic tools‌ are most often used to study these problems.‌ The dual point of view is obtained thanks‌ to Motzkin's theorem of the alternative, rather than‌ Gordan's. The Julia code isf_jl (§ 7.1.11 and‌ 55) has been written to assess the‌ efficiency of the algorithms proposed to enumerate the‌ chambers of an affine arrangement. This code is‌ also able to list the bounded chambers of‌ the arrangement, when this one is in linear‌ general position, which is a useful information in‌ the analysis of hypergeometric functions, as well as‌ in cosmology and particle physics.

8.2 Research axis‌ 2: A posteriori error control, adaptivity, and safe‌ and correct programming

Participants: François Clément, Zhaonan‌ Dong, Alexandre Ern, Nicolas Hugot,‌ Lukas Renelt, Martin Vohralík, Zuodong Wang‌, Benjamin Zurich.

A posteriori error estimates‌ for the wave equation

Participants: Zhaonan Dong,‌ Alexandre Ern, Zuodong Wang.

A posteriori‌ error estimates for the wave equation are quite‌ challenging (in particular, because of the lack of‌ a well established inf-sup stability framework), and results‌ in the literature are still quite scarce. In‌ 21, we considered damped energy-norm error estimates‌ and were able to establish both upper and‌ lower bounds on the error, up to some‌ higher-order terms, under the assumptions of smooth enough‌ and compactly supported sources away from the origin.‌

In 51, we establish rigorous a posteriori‌ error bounds for a space-time finite element method‌ of arbitrary order discretising linear wave problems in‌ second order formulation. The method combines standard finite‌ elements in space and continuous piecewise polynomials in‌ time with an upwind discontinuous Galerkin-type approximation for‌ the second temporal derivative. The proposed scheme accepts‌ dynamic mesh modification, as required by space-time adaptive‌ algorithms, resulting in a discontinuous temporal discretisation when‌ mesh changes occur. We prove a posteriori error‌ bounds in the ${L}^{\infty} (L^{2‌})$ -norm, using carefully designed temporal and spatial‌ reconstructions; explicit control on the constants (including the‌ spatial and temporal orders of the method) in‌ those error bounds is shown. The convergence behavior‌ of an error estimator is verified numerically, also‌ taking into account the effect of the mesh‌ change. A space-time adaptive algorithm is proposed and‌ tested numerically. Some of the technical results have‌ been established in the early work 52.‌

Figure 3.a — Figure 3: Visualization‌ of dynamic mesh modification of the test case‌ with space-time adaptive algorithm; $q = 2$ ;‌ $p = 1$ (top) and $p = 2‌$ (bottom); $T$ equals 0, $0.1$ ,‌ $0.3$ and $0.5$ from‌ left to right.

Figure 3.b — Figure 3: Visualization‌ of dynamic mesh modification of the test case‌ with space-time adaptive algorithm; $q = 2$ ;‌ $p = 1$ (top) and $p = 2‌$ (bottom); $T$ equals 0, $0.1$ ,‌ $0.3$ and $0.5$ from‌ left to right.

Book chapter on a posteriori‌ error estimates

Participants: Martin Vohralík.

The book‌ chapter 37 (M. Vohralík with the colleague S.‌ Yousef from IFP Energies Nouvelles, 130 pages)‌ makes an introduction to the theory of a posteriori error estimates, allowing‌ to give bounds on‌ the error between the‌‌ unknown solution of a partial differential equation and‌ its numerical approximation. Any‌ lowest-order locally conservative method‌‌ of the finite volume type is considered and‌ general polytopal meshes are‌ treated. The whole range‌‌ between the Laplace model problem and complex multiphase‌ multicomponent flows in porous‌ media is addressed. Inexpensive‌‌ implementation and evaluation of the estimators is detailed,‌ and the use of‌ iterative linearization and algebraic‌‌ solvers is taken into account and employed in‌ order to desing adaptivity.‌ Numerical experiments on academic‌‌ benchmarks as well as on real-life underground porous‌ media problems in two‌ and three space dimensions‌‌ illustrate the performance of the derived methodology. Figure‌ 4 showcases the performance‌ of the discussed estimates‌‌ on polygonal meshes in two space dimensions.

Figure 4.a — Figure 4:‌ Energy error (most left)‌ and a posteriori error‌‌ estimates on a triangular submesh (left), on polygons‌ using the MFE element‌ matrices (simpler evaluation, middle),‌‌ and on polygons using the built-in polytopal scheme‌ element matrices (simplest evaluation,‌ right). Full doman (top‌‌ line) and zoom (bottom line)

Figure 4.b — Figure 4:‌ Energy error (most left)‌ and a posteriori error‌‌ estimates on a triangular submesh (left), on polygons‌ using the MFE element‌ matrices (simpler evaluation, middle),‌‌ and on polygons using the built-in polytopal scheme‌ element matrices (simplest evaluation,‌ right). Full doman (top‌‌ line) and zoom (bottom line)

$h p$ -a‌ posteriori error estimates for‌ nonconforming methods

Participants: Zhaonan‌‌ Dong, Alexandre Ern.

In 48,‌ we derived $h p‌$ -a posteriori error estimates‌‌ for the nonconforming approximation of Maxwell's equations in‌ second-order form. The main‌ idea is the use‌‌ of local Helmholtz decompositions combined with a partition-of-unity‌ technique to bound the‌ nonconformity error. Another salient‌‌ contribution of independent interest is the devising and‌ $h p$ -analysis of‌ a H(curl)-conforming reconstruction operator‌‌ from piecewsie polynomial fields on vertex-based patches in‌ simplicial meshes.

Rocq formalizations‌

Participants: François Clément.‌‌

In 43, we formalize a finite element‌ as a record in‌ the Rocq proof assistant‌‌ (formerly known as Coq) with both values and‌ proofs of validity, including‌ the main one called‌‌ unisolvence. We then instantiate this record with the‌ most popular and useful,‌ the simplicial Lagrange finite‌‌ elements for evenly distributed nodes, for any dimension‌ and any polynomial degree.‌ These proofs require many‌‌ results (definitions, lemmas, canonical structures) about finite families,‌ affine spaces, multidimensional polynomials,‌ in the context of‌‌ finite of infinite-dimensional spaces.

In 42, we‌ formalize operators and proofs‌ of properties about relations‌‌ and orders, and we especially focus on monomial‌ orders, that are total‌ orders compatible with the‌‌ monoid operation. For the sake of genericity, we‌ formalize the grading of‌ an order, a high-level‌‌ operator that transforms a binary relation into another‌ one, and we prove‌ that grading an order‌‌ preserves many of its properties, such as the‌ monomial order property. The‌ graded symmetric lexicographic order‌‌ is chosen to order multi-indices, and thus the‌ nodes of simplicial Lagrange‌ finite elements.

See also‌‌ the "News of the Year" about software rocq-num-analysis‌ (Section 7.1.4).

8.3‌ Research axis 3: Applications‌‌ to environment and energy‌

Participants: Alexandre Ern, Abbas Kabalan, Michel‌ Kern, Ibtissem Lannabi, Simon Legrand,‌ Romain Mottier, Géraldine Pichot, Martin Vohralík‌, Raphaël Zanella, Daniel Zegarra Vasquez.‌

Flow simulations in fracture networks

Participants: Zhaonan Dong‌, Michel Kern, Géraldine Pichot.

The‌ computational costs of flow simulations in the underground‌ could be prohibitive. To save computational time and‌ resources, adaptive mesh strategies are very promising. They‌ require two main ingredients: discretization methods capable of‌ supporting general elements, as Hybrid High-Order (HHO) and‌ Discontinuous Galerkin (DG) methods, and a posteriori error‌ estimates to drive the mesh adaptivity. As part‌ of the STEERS ANR project, we have tested‌ the combination of HHO/DG and residual-type estimators in‌ the context of flow simulations in fractures network.‌ We have shown that the initial coarsening stage‌ and the adaptivity loop make it possible to‌ reduce significantly the number of unknowns without compromising‌ the accuracy of the flow simulations (an example‌ is shown on figures 5 and 6).‌ A paper is in preparation.

Figure 5.a — Figure‌ 5: Fracture network and hydraulic head

Figure 5.b — Figure‌ 5: Fracture network and hydraulic head

Figure 6.a — Figure 6: (left) Initial fine mesh,‌ (right) adapted mesh obtained after 14 levels of‌ coarsening

Figure 6.b — Figure 6: (left) Initial fine mesh,‌ (right) adapted mesh obtained after 14 levels of‌ coarsening

Flow through fractured porous media

Participants: Michel‌ Kern, Simon Legrand, Géraldine Pichot,‌ Raphaël Zanella, Daniel Zegarra Vasquez.

The‌ Ph.D. thesis of Daniel Zegarra Vasquez , concerned‌ with large scale simulation of flow in fractured‌ porous media, was defended in May 2025. The‌ main results contained in the thesis were:

A‌ thorough mathematical and numerical analysis of the coupled‌ mixed-dimensional system of PDEs, and its approximation by‌ mixed-hybrid finite elements 64;
A comparison of‌ different algebraic solvers applied to the resulting large‌ and highly ill-conditioned linear system, showing that the‌ black-box solvers cannot tackle this problem 65;‌
A demonstration that the two-level domain decomposition preconditioner‌ HPDDM-GenEO, developped by the Alpines team 69‌, 70 consistently outperformed all other methods, as‌ can be seen on Figure 7 62.‌

Figure 7: Comparison of the performance‌ of HPDDM-GenEO with other preconditioners for solving‌ flow in fractured porous media

As an example,‌ solving the linear system of size $243 \times‌ 10^{6}$ for a network containing 696k fractures‌ takes only 4 minutes and 49 iterations in‌ parallel with 6825 MPI processes, using GMRES preconditioned‌ with HPDDM. The linear systems are generated‌ witht the nef-flow-fpm (§ 7.1.6) software, and‌ are solved with the PETSc-HPDDM. All‌ simulations were run on the TGCC (GENCI /‌ CEA).

Multiphase flow

Participants: Michel Kern.

Etienne Ahusborde (Université de Pau)‌ and Michel Kern formed‌ a team that participated‌‌ in The 11th Society of Petroleum Engineers. Comparative‌ Solution Project. The‌ goal was to compare‌‌ simulation codes for CO2 sequestration on realistic test‌ cases. Our team presented‌ results for all three‌‌ test cases that were in line with the‌ majority of other groups,‌ see Figures 8 and‌‌ 9. A joint paper synthesized the results‌ submitted by all the‌ participating groups for all‌‌ three models 33.

Figure 8: Comparison of all results‌ (pressure field) for SPE11‌ B model (from 33‌‌)

Figure 9: CO2 molar fraction after 500‌ years, for SPE11 C‌ model (from SPE CSP11‌‌ results)

Wave propagation in geophysical media

Participants:‌ Alexandre Ern, Romain‌ Mottier.

Within the‌‌ Ph.D. thesis of Romain Mottier, defended in July‌ 2025, 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.‌ An illustration is shown‌ in Figure 10.‌‌ The sedimentary bassin is located to the left‌ of the hill and‌ stores a significant part‌‌ of the elastic energy before gradually releasing it‌ into the atmosphere. More‌ comprehensive numerical results can‌‌ be found in 66.

Figure‌ 10: Wave propagation‌‌ in a sedimentary bassin.

Data assimilation

Participants: Alexandre‌ Ern.

Our work‌ on data assimilation was‌‌ pursued this year by addressing the wave equation.‌ Our main 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. The step‌ forward with respect to‌ our previous work on‌‌ the heat equation is that the conditional stability‌ of the wave equation‌ is more delicate. Moreover,‌‌ the stabilization strategy had to be revised in‌ order to cope with‌ the higher-order time derivatives.‌‌

8.4 Research axis 4: PDE and numerical analysis‌ foundations

Participants: Zhaonan Dong‌, Alexandre Ern,‌‌ Jean-Luc Guermond, Martin Vohralík.

Stable local‌ commuting projectors

Participants: Alexandre‌ Ern, Martin Vohralík‌‌.

A long-standing question in the literature has‌ been to construct interpolation‌ operators on the canonical‌‌ finite element spaces (Lagrange, Nédélec, Raviart–Thomas and discontinuous‌ finite element spaces on‌ simplicial meshes) that enjoy‌‌ three key properties: (i) They are locally defined;‌ (ii) They are ${L‌}^{2}$ -stable; (iii) They‌‌ commute with the differential operators grad, curl and‌ div. In 60,‌ we brought a positive‌‌ answer to this quest. Our main construction is‌ agnostic to boundary conditions,‌ but we also showed‌‌ how to modify the construction so as to‌ preserve the corresponding homogeneous‌ boundary conditions (zero trace‌‌ for Lagrange elements, zero‌ tangential trace for the Nédélec elements, and zero‌ normal trace for the Raviart–Thomas elements). Our ${L‌}^{2}$ -stability proof hinges on a result of‌ independent interest, published in 30, namely discrete‌ Poincaré inequalities, specifically on stars associated with the‌ various geometric entities of the mesh (vertices, edges‌ and faces).

Quasi-interpolation with computable error bounds

Participants:‌ Martin Vohralík.

In 22, we design‌ a quasi-interpolation operator from the Sobolev space ${H‌}_{0}^{1} (Ω)$ to its finite-dimensional‌ finite element subspace formed by piecewise polynomials on‌ a simplicial mesh with a computable approximation constant.‌ The operator 1) is defined on the entire‌ $H_{0}^{1} ( Ω)$ , no‌ additional regularity is needed; 2) allows for an‌ arbitrary polynomial degree; 3) works in any space‌ dimension; 4) is defined locally, in vertex patches‌ of mesh elements; 5) yields optimal estimates for‌ both the $H^{1}$ seminorm and the ${L‌}^{2}$ norm error; 6) gives a computable constant‌ for both the ${H}^{1}$ seminorm and the‌ $L^{2}$ norm error; 7) leads to the‌ equivalence of global-best and local-best errors; 8) possesses‌ the projection property. Its construction follows the so-called‌ potential reconstruction from a posteriori error analysis. Numerical‌ experiments illustrate that our quasi-interpolation operator systematically gives‌ the correct convergence rates in both the ${H‌}^{1}$ seminorm and the $L^{2}$ norm and‌ its certified overestimation factor is rather sharp and‌ stable in all tested situations.

Maxwell's equations

Participants:‌ Alexandre Ern, Jean-Luc Guermond.

In 20‌, 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. As a further development, we also‌ established in 19 a similar result for the‌ discontinuous Galerkin (dG)approximation, this time involving an additional‌ high-order perturbation related to the approximated flux. Additionally,‌ we devised a novel residual-based a posteriori error‌ analysis for the Maxwell problem in the frequency‌ domain.

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 with constant coefficients last year, and generalized it‌ to the case of variable coefficients in 28‌. The analysis is quite challenging, and relies‌ upon a duality argument and a deflated inf-sup‌ condition. An important consequence of this result, presented‌ in 58, is the convergence proof for‌ the approximation of the time-dependent Maxwell's equations in‌ first-order form with minimal regularity assumptions, using a‌ dG method with upwinding in space and a‌ third-order or fourth-order, explicit Runge–Kutta method in time.‌ In particular, our convergence result holds for a Sobolev regularity index $s‌ > 0$ , whereas‌ all the previous results‌‌ in the literature required $s > \frac{1}{2‌}$ . One salient contribution‌ in 58 is to‌‌ emphasize the central role played by involutions (Gauss‌ laws in the present‌ case) in the analysis.‌‌

Sign-changing elliptic PDEs

Participants: Alexandre Ern.

Sign-changing‌ elliptic PDEs are encountered‌ in metamaterials leading to‌‌ propagation of waves in ways that are unknown‌ from naturally ocurring materials.‌ The simplest model is‌‌ that of a second-order diffusive PDE posed on‌ two sub-domains separated by‌ an interface, with a‌‌ positive diffusion coefficient in one sub-domain and a‌ negative coefficient in the‌ other. The discretization of‌‌ such problems is quite challenging. The standard Galerkin‌ discretization is stable only‌ under rather strong assumptions‌‌ on the mesh, essentially hinging on some symmetry‌ properties that are difficult‌ to satisfy in practice.‌‌ Our main contribution is a novel discretization method‌ that is stable on‌ general shape-regular meshes that‌‌ are fitted to the interface. The key idea‌ is to employ a‌ stabilized primal-dual approach. Thus,‌‌ the price to pay to gain better stability‌ is to solve a‌ saddle-point problem. The numerical‌‌ analysis along simulation results on challenging problems are‌ reported in 18.‌ An illustration is presented‌‌ in Figure 11. We compare the relative‌ errors obtained using the‌ standard Galerkin method (red‌‌ curves) and the present method (blue curves). When‌ the mesh is symmetric,‌ the standard Galerkin method‌‌ remains well behaved, even when the diffusion coefficients‌ approach the critical contrast.‌ Instead, the Galerkin method‌‌ becomes unstable as soon as the symmetry is‌ lost, whereas the present‌ method exhibits robust behavior.‌‌

Figure 11: Relative error‌ for the standard Galerkin‌ approach (red) and the‌‌ present approach (blue) for a cavity with diffusion‌ coefficients approaching the critical‌ contrast. Left: unstructured mesh;‌‌ right: symmetric mesh.

Model-order reduction with geometric variability‌

Participants: Alexandre Ern,‌ Abbas Kabalan.

One‌‌ important topic has been the development of reduced-order‌ methods with geometric variability‌ in the context of‌‌ aeronautics simulations. This topic is explored in the‌ Ph.D. thesis of Abbas‌ Kabalan , defended this‌‌ year and supported by SafranTech. The main‌ achievements are elasticity-based morphing‌ techniques 31 and an‌‌ optimization method to improve the compressibility of snapshots‌ 63. While the‌ first approach constructs a‌‌ morphing for each sample individually and only considers‌ the geometry, the second‌ approach constructs the morphings‌‌ collectively for all the samples in the datasets‌ and considers the variability‌ in the geometry and‌‌ also in a specific output. Figure 12 illustrates‌ the Mach field (the‌ output) with different geometry‌‌ and shock structures from three samples of the‌ VKI dataset consisting of‌ 2D compressible steady-state RANS‌‌ simulations of flow fields around turbine blades.

Figure 12.a — Figure 12: Three samples‌ from the VKI turbine blade dataset: Mach field‌ profiles exhibiting different geometry and different shock structures.‌

Figure 12.b — Figure 12: Three samples‌ from the VKI turbine blade dataset: Mach field‌ profiles exhibiting different geometry and different shock structures.‌

Inviscid total variation and Bingham minimizers

Participants: Alexandre‌ Ern.

Inviscid total variation and Bingham minimizers‌ are important limit problems to understand the flow‌ of visco-plastic materials. The total variation minimization problem‌ serves as a simpler, scalar-valued, problem to tackle‌ the more difficult setting of Bingham flows. In‌ 16, we obtained new regularity results on‌ the minimizers of such problems for $H^{1‌}$ -data. We also showed how homogeneous Dirichlet conditions‌ on the viscous problem lead in the vanishing‌ viscosity limit to relaxed boundary conditions of frictional‌ type.

A hypocoercivity-exploiting stabilised finite element method for‌ Kolmogorov equation

Participants: Zhaonan Dong.

In 49‌, we are concerned with discretisations of the‌ classical Kolmogorov equation by a standard space-time discontinuous‌ Galerkin method. Kolmogorov equation serves as simple, yet‌ rich enough in the present context, model problem‌ for a wide range of kinetic-type equations: although‌ it involves diffusion in one of the two‌ spatial dimensions only, the combined nature of the‌ first order transport/drift term and the degenerate diffusion‌ are sufficient to ‘propagate dissipation’ across the spatial‌ domain in its entirety. This is a manifestation‌ of the celebrated concept of hypocoercivity, a term‌ coined and studied extensively by Villani. We show‌ that the standard, classical, spacetime discontinuous Galerkin method,‌ admits a corresponding hypocoercivity property at the discrete‌ level, asymptotically for large times. To the best‌ of our knowledge, this is the first result‌ of this kind for any standard Galerkin scheme.‌ This property is shown by proving one part‌ of a discrete inf-sup-type stability result for the‌ method in a family of norms dictated by‌ a modified scalar product motivated by the theory‌ in Villani. This family of norms contains the‌ full gradient of the numerical solution, thereby allowing‌ for a full spectral gap/Poincaré-type inequality at the‌ discrete level, thus, showcasing a subtle, discretisation-parameter-dependent, numerical‌ hypocoercivity property. Further, we show that the space-time‌ discontinuous Galerkin method is inf sup stable in‌ the family of norms containing the full gradient‌ of the numerical solution, which may be a‌ result of independent interest.

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 SafranTech accompanying the‌ PhD thesis of Abbas Kabalan (co-supervised with V.‌ Ehrlacher), ended on 31/12/2025.
Two-part contract with CEA‌ accompanying the PhD thesis of Romain Mottier, ended‌ on 31/12/2025.
Two-part contract with IFP Energies Nouvelles‌ accompanying the post-doc of Ibtissem Lannabi.
Two-part contract‌ with Total Energies developing adaptive stopping criteria for‌ linear and nonlinear solvers as well as adaptive‌ regularization for geological sequestration of CO2 (Arthur Moncorgé).‌

10 Partnerships and cooperations

10.1 International initiatives

10.1.1‌ Participation in other International Programs

RANPDEs

Participants: Lukas Renelt, Martin Vohralík‌, Benjamin Zurich.‌

Title: Robust adaptivity for‌‌ nonlinear partial differential equations
Partner Institution:
University of‌ Bonn, Germany (Gregor Gantner)‌
Date/Duration: 2025–2028
Additionnal info/keywords:‌‌
ANR–DFG international grant project. The goal‌ is to prove contraction‌ on each step and‌‌ cost-optimality of adaptive algorithms for model nonlinear problems,‌ necessarily relying on an‌ interplay between analysis of‌‌ partial differential equations, numerical analysis, and numerical linear‌ algebra.

AdaptPMod

Participants: Martin‌ Vohralík.

Title: Model‌‌ adaptivity in the numerical simulation of porous media‌ flows
Partner Institution:
University‌ of Hasselt, Belgium (Iuliu‌‌ Sorin Pop)
Date/Duration: 2025–2028
Additionnal info/keywords:
FWO international‌ grant project. The goal‌ is to design automatic‌‌ switching between a hierarchy of models used in‌ numerical simulations of porous‌ media flows (Darcy, Richards,‌‌ two-phase, compositional, ...).

10.2 International research visitors

10.2.1‌ Visits of international scientists‌

Other international visits to‌‌ the team

Jean-Luc Guermond

Status:
Professor
Institution of‌ origin:
Texas A&M University‌
Country:
USA
Dates:
05/05‌‌ to 04/07/2025
Context of the visit:
Collaboration on‌ scientific projects related to‌ involution-preserving discretizations
Mobility program/type‌‌ of mobility:
two-month invited Professor position, one month‌ funded by University Paris-Est‌ and one month by‌‌ INRIA Paris

Buyang Li

Status:
Professor
Institution of‌ origin:
Hong Kong Polytechnic‌ University
Country:
China
Dates:‌‌
11/06 to 10/07/2025
Context of the visit:
Collaboration‌ on scientific projects related‌ to numerical methods for‌‌ the fluid structure interaction problems
Mobility program/type of‌ mobility:
one-month invited Professor‌ position by INRIA Paris‌‌

Bernardo Cockburn

Status:
Professor
Institution of origin:
University‌ of Minnesota
Country:
USA‌
Dates:
01/11 to 30/11/2025‌‌
Context of the visit:
Collaboration on scientific projects‌ related to HDG/HHO schemes‌
Mobility program/type of mobility:‌‌
One-month invited Professor position funded by Labex Bezout‌

10.3 National initiatives

Participants:‌ Zhaonan Dong, Alexandre‌‌ Ern, Michel Kern, Géraldine Pichot.‌

Alexandre Ern is the‌ Director of the CNRS‌‌ Thematic Network on Earth and Energies since 01/01/2024.‌ The goal of the‌ network is to foster‌‌ collaborations on mathematical topics applied to Earth and‌ Environment sciences. In 2025,‌ the Thematic Network partially‌‌ supported various conferences and workshops held in France‌ on topics related to‌ the mathematical modelling and‌‌ numerical simulation of environment-related problems.

Zhaonan Dong ,‌ Michel Kern , and‌ Géraldine Pichot were awarded‌‌ an ANR PRME grant, called STEERS (Space-Time adaptivE‌ mEthods for subsuRface flow‌ Simulations, 2025-2028). The goal‌‌ is to develop new robust and efficient methods‌ for the simulation of‌ subsurface flow, using a‌‌ combined Hybrid High-Order / Discontinuous Galerkin (HHO/DG) method‌ on agglomerated meshes, and‌ to speed up the‌‌ computations using new space-time adaptive algorithms steered by‌ a posteriori error estimates,‌ aiming at guaranteeing solutions‌‌ to a given user accuracy. The long term‌ objective is to develop‌ robust, accurate and efficient‌‌ numerical methods, algorithms and an open-source library for‌ the simulations of CO2‌ storage in deep geological‌‌ formations.

11 Dissemination

11.1 Promoting scientific activities

11.1.1‌ Scientific events: organisation

General‌ chair, scientific chair

Alexandre‌‌ Ern was the French‌ co-organizer of the Indo-French workshop on Innovative Numerical‌ Methods for Modern Engineering Problems held at IIT‌ Roorkee (India) on 06–10/01/2025.

Martin Vohralík was a‌ co-organizer of the workshop Frontiers in Numerical Methods‌ for Partial Differential Equations held at EPFL Lausanne‌ (Switzerland), October 08–10, 2025.

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

Member of the organizing committees‌

Alexandre Ern is a member of the Scientific‌ Committee of the European Finite Element Fair.‌ The 2025 Fair was held at SISSA (University‌ of Tieste, Italy) in May 2025.

11.1.2 Scientific‌ events: selection

Member of the conference program committees‌

Alexandre Ern is since September 2025 member of‌ the ENUMATH Program Committee. Martin Vohralík is a‌ member of the Scientific Committee of EMS School‌ on Mathematical Modelling, Numerical Analysis and Scientific Computing‌.

11.1.3 Journal

Member of the editorial boards‌

Alexandre Ern is co-Editor-in-Chief of IMA Journal of‌ Numerical Analysis (IMAJNA) since 01/01/2024. Moreover, he continues‌ to serve as Associate Editor for SIAM Journal‌ on Scientific Computing (SISC), ESAIM Mathematical Modeling and‌ Numerical Analysis (M2AN), Journal of Scientific Computing (JOMP),‌ and Computational Methods in Applied Mathematics (CMAM). Martin‌ Vohralík is a member of the editorial boards‌ of Acta Polytechnica, Applications of Mathematics,‌ and Computational Geosciences.

Reviewer - reviewing activities‌

Alexandre Ern , Zhaonan Dong , Michel Kern‌ , and Martin Vohralík served as reviewers for‌ various articles in numerical analysis and computational PDEs.‌

11.1.4 Invited talks

Alexandre Ern delivered one of‌ the two keynote lectures at the Scientific Day‌ on Applied Mathematics organized by EdF Lab at‌ Saclay in October 2025.

Géraldine Pichot gave a‌ keynote presentation in the Joint Brazil-Chile-Inria MS on‌ Innovative Numerical Methods for Fluids, at the 23rd‌ IACM Computational Fluids Conference at Santiago de Chile,‌ in March 2025.

Géraldine Pichot was an invited‌ speaker at the Workshop on "Interfaces and Unfitted‌ Discretization Methods" at the Mittag-Leffler Institute (Sweeden) in‌ December 2025.

Martin Vohralík was an invited speaker‌ at the Dutch–Flemish scientific computing society annual meeting‌, Woudschoten, The Netherlands, September 2025.

11.1.5 Leadership‌ within the scientific community

Michel Kern is a‌ member of the Scientific Board of ORAP (Organisation‌ Associative du Parallélisme).

Michel Kern serves on the‌ steering committee of GDR HydroGEMM (“Hydrogène du sous-sol:‌ étude intégrée de la Genèse ...à la Modélisation‌ Mathématique”)

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

Martin Vohralík serves‌ in 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 for the‌ Juelich Supercomputing Centre in Germany.

Michel Kern is‌ an external member of the board of the‌ École Doctorale Galilée at University Sorbonne Paris-Nord.

Michel‌ Kern is a member of the selection committe‌ of the SIAM John von Neumann Prize.

Martin Vohralík serves as‌ a member of the‌ competition and certification committee‌‌ at the Institute of Mathematics, Czech Academy of‌ Sciences, since 2024.‌

11.1.7 Research administration

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.

Géraldine Pichot‌ set up an Inria‌ stand at the 14ème‌‌ Forum Entreprises et Mathématiques.

Géraldine Pichot is‌ a member of the‌ Conseil du Département Mathématiques‌‌ Appliquées Polytech Lyon.

11.2 Teaching - Supervision -‌ Juries - Educational and‌ pedagogical outreach

11.2.1 Supervision‌‌

Ph.D. defended: Abbas Kabalan , Model-order reduction for‌ non-parametrized geometric variability with‌ application to aeronautics simulations,‌‌ defended on 17/12/2025, supervised by Virginie Ehrlacher (ENPC),‌ Alexandre Ern , and‌ Fabien Casenave (SafranTech‌‌).
Ph.D. defended: Romain Mottier , Hybrid high-order‌ methods for the numerical‌ simulation of elasto-acoustic wave‌‌ propagation, defended on 23/07/2025, supervised by Alexandre Ern‌ and Laurent Guillot (CEA).‌
Ph.D. defended: Baptiste Plaquevent-Jourdain‌‌ , A robust linearization method for complementarity problems‌ - a detour through‌ hyperplane arrangements, defended on‌‌ 16/07/2025, supervised by Jean-Pierre Dussault (Univ. de Sherbrooke,‌ Canada) and Jean Charles‌ Gilbert .
Ph.D. defended:‌‌ Zuodong Wang , Finite element methods for hyperbolic‌ and degenerate parabolic problems,‌ defended on 23/09/2025, supervised‌‌ by Alexandre Ern and Zhaonan Dong .
Ph.D.‌ defended: Daniel Zegarra Vasquez‌ , Efficient numerical simulation‌‌ of single-phase flow in three-dimensional fractured porous media,‌ defended on 27/05/2025, supervised‌ by Geraldine Pichot ,‌‌ Michel Kern , and Martin Vohralík .
Ph.D.‌ in progress: Nicolas Hugot‌ , A posteriori error‌‌ estimates for the wave equation, started November 2023,‌ supervised by Alexandre Imperiale‌ (CEA LIST)‌‌ and Martin Vohralík.
Ph.D. in progress: Clément Maradei‌ , Inexpensive $p$ -uniform‌ iterative solvers, started October‌‌ 2023, supervised by Zhaonan Dong and Martin Vohralík‌ .
Ph.D. in progress:‌ Bahaa Eddine Sidi Hida‌‌ , Preconditionning and coupling for HHO methods, started‌ on 01/01/2025, supervised by‌ Alexandre Ern and Pierre‌‌ Jolivet (Sorbonne University).
Ph.D. in progress: Benjamin Zurich‌ , Mesh and solvers‌ adaptivity for nonlinear partial‌‌ differential equations: contraction and optimality, started June 2025,‌ supervised by André Harnist‌ (UTC Compiègne) and Martin‌‌ Vohralík.

11.2.2 Juries

Alexandre Ern was a member‌ of the HdR defense‌ committee of Laurent Orgogozo‌‌ (University of Toulouse, 09/2025) and chaired the Ph.D.‌ defense committee for Simone‌ Pescuma (IPP, ENSTA, 11/2025).‌‌
Martin Vohralík was the chair of the HdR‌ defense committee of Maxime‌ Breden (Ecole Polytechnique, 06/2025),‌‌ the chair of the Ph.D. defense committee of‌ Hugo Dornier (Ecole Polytechnique,‌ 12/2025), and a referee‌‌ of the Ph.D. defense committee of Lukas Renelt‌ (University of Münster, Germany,‌ 01/2025).

11.2.3 Educational and‌‌ pedagogical outreach

Master (M2): Alexandre Ern , Discontinuous‌ Galerkin methods, 20h, M2,‌ Sorbonne University, France.
Master‌‌ (M1): Alexandre Ern ,‌ Finite Elements, 15h, M1, ENPC, France.
Master (M2):‌ Alexandre Ern , Hyperbolic equations, 6h, M2, Sorbonne‌ University, France.
Master (M2): Michel Kern , Simulation‌ of subsurface flow, 20h, M2, Paris-Saclay University.
Master‌ (M1): Michel Kern , Iterative methods for linear‌ systems, 20h, M1, Sorbonne Paris-Nord Univesrité, France.
Master‌ (M1): Martin Vohralík , Advanced finite elements, 21h,‌ M1, ENSTA (Ecole nationale supérieure de techniques avancées),‌ Paris-Saclay, France.

11.3 Popularization

11.3.1 Participation in Live‌ events

Michel Kern , Geraldine Pichot , and‌ Martin Vohralík participated in the internship week for‌ middle school and high schools students (stages d'observation‌ pour les classes de 3ème et de seconde).‌

Michel Kern gave talks to high school students‌ (Classe de seconde) on “Numerical simulations and subsurface‌ flow” in the framework of the Chiche (“1‌ scientifique, 1 classe : chiche !”) program.

12‌ Scientific production

12.1 Major publications

1 articleL.‌Laila Amir and M.Michel Kern. Preconditioning‌ a coupled model for reactive transport in porous‌ media.International Journal of Numerical Analysis and‌ Modeling1612019, 18-48HAL
2‌ articleS.Sylvie Boldo, F.François Clément‌, F.Florian Faissole, V.Vincent Martin‌ and M.Micaela Mayero. A Coq Formalization‌ of Lebesgue Integration of Nonnegative Functions.Journal‌ of Automated Reasoning662022, 175–213HAL‌DOI
3 inproceedingsS.Sylvie Boldo, F.‌François Clément, F.Florian Faissole, V.‌Vincent Martin and M.Micaela Mayero. A‌ Coq formal proof of the Lax–Milgram theorem.‌6th ACM SIGPLAN Conference on Certified Programs and‌ ProofsParis, FranceJanuary 2017HAL DOI
4‌ articleE.Eric Cancès, G.Geneviève Dusson‌, Y.Yvon Maday, B.Benjamin Stamm‌ and M.Martin Vohralík. Guaranteed and robust‌ a posteriori bounds for Laplace eigenvalues and eigenvectors:‌ conforming approximations.SIAM Journal on Numerical Analysis‌555September 2017, 2228-2254HAL DOI‌
5 articleA.Andrea Cangiani, Z.Zhaonan‌ Dong and E. H.Emmanuil H Georgoulis.‌ $h p$ -Version discontinuous Galerkin methods on essentially‌ arbitrarily-shaped elements.Mathematics of Computation91333‌January 2022, 1-35HAL DOI
6 article‌A.Andrea Cangiani, Z.Zhaonan Dong and‌ E. H.Emmanuil H Georgoulis. A posteriori‌ error estimates for discontinuous Galerkin methods on polygonal‌ and polyhedral meshes.SIAM Journal on Numerical‌ Analysis6152023, 2352--2380HAL DOI‌
7 articleD. A.Daniele A. Di Pietro‌ and A.Alexandre Ern. A hybrid high-order‌ locking-free method for linear elasticity on general meshes‌.Comput. Methods Appl. Mech. Engrg.2832015‌, 1--21URL: http://dx.doi.org/10.1016/j.cma.2014.09.009DOI
8 articleJ.-P.‌Jean-Pierre Dussault and J. C.Jean Charles Gilbert‌. Exact computation of an error bound for‌ the balanced linear complementarity problem with unique solution‌.Mathematical Programming1991-22023, 1221-1238‌HAL DOI
9 articleA.Alexandre Ern,‌ T.Thirupathi Gudi, I.Iain Smears and M.Martin Vohralík.‌ Equivalence of local-and global-best‌ approximations, a simple stable‌‌ local commuting projector, and optimal $h p$ approximation‌ estimates in $H$ (div)‌.IMA Journal of‌‌ Numerical Analysis422April 2022, 1023-1049‌HAL DOI
10 article‌A.Alexandre Ern and‌‌ J.-L.Jean-Luc Guermond. Spectral correctness of the‌ discontinuous Galerkin approximation of‌ the first-order form of‌‌ Maxwell's equations with discontinuous coefficients.SIAM Journal‌ on Numerical Analysis63‌22025, 661-684‌‌HAL DOI
11 articleA.Alexandre Ern,‌ F.Florent Hédin,‌ G.Géraldine Pichot and‌‌ N.Nicolas Pignet. Hybrid high-order methods for‌ flow simulations in extremely‌ large discrete fracture networks‌‌.SMAI Journal of Computational Mathematics82022‌, 375-398HAL DOI‌
12 articleA.Alexandre‌‌ Ern and M.Martin Vohralík. Polynomial-degree-robust a‌ posteriori estimates in a‌ unified setting for conforming,‌‌ nonconforming, discontinuous Galerkin, and mixed discretizations.SIAM‌ Journal on Numerical Analysis‌532April 2015‌‌, 1058-1081HAL DOI
13 articleA.Alexandre‌ Ern and M.Martin‌ Vohralík. Stable broken‌‌ H1 and H(div) polynomial extensions for polynomial-degree-robust potential‌ and flux reconstruction in‌ three space dimensions.‌‌Mathematics of Computation89322March 2020,‌ 551-594HAL DOI
14‌ articleG.Géraldine Pichot‌‌, J.Jocelyne Erhel and J.-R.Jean-Raynald De‌ Dreuzy. A generalized‌ mixed hybrid mortar method‌‌ for solving flow in stochastic discrete fracture networks‌.SIAM J. Sci.‌ Comput.3412012‌‌, B86--B105URL: http://dx.doi.org/10.1137/100804383DOI

12.2 Publications of‌ the year

International journals‌

15 articleM.Manuela‌‌ Bastidas Olivares, A.Akram Beni Hamad,‌ M.Martin Vohralík and‌ I.Ivan Yotov.‌‌ A posteriori algebraic error estimates and nonoverlapping domain‌ decomposition in mixed formulations:‌ energy coarse grid balancing,‌‌ local mass conservation on each step, and line‌ search.Computer Methods‌ in Applied Mechanics and‌‌ Engineering444September 2025, 118090HAL DOI‌
16 articleF.François‌ Bouchut, C.Carsten‌‌ Carstensen and A.Alexandre Ern. $H^{1‌}$ regularity of the minimizers‌ for the inviscid total‌‌ variation and Bingham fluid problems for $H^{1‌}$ data.Nonlinear Analysis:‌ Theory, Methods and Applications‌‌258March 2025, 113809HAL DOI back‌ to text
17 article‌E.Erik Burman,‌‌ G.Guillaume Delay and A.Alexandre Ern.‌ The unique continuation problem‌ for the wave equation‌‌ discretized with a high-order space-time nonconforming method.‌Numerische Mathematik1574‌2025, 1259-1284HAL‌‌DOI back to text
18 articleE.Erik‌ Burman, A.Alexandre‌ Ern and J.Janosch‌‌ Preuss. A stabilized hybridized Nitsche method for‌ sign-changing elliptic PDEs.‌The Scientific World Journal‌‌35142025, 2977-3009HAL DOI back‌ to text
19 article‌T.Théophile Chaumont-Frelet and‌‌ A.Alexandre Ern. A priori and a‌ posteriori analysis of the‌ discontinuous Galerkin approximation of‌‌ the time-harmonic Maxwell's equations under minimal regularity assumptions‌.Mathematics of Computation‌2025HAL DOI back‌‌ to text
20 article‌T.Théophile Chaumont-Frelet and A.Alexandre Ern.‌ Asymptotic optimality of the edge finite element approximation‌ of the time-harmonic Maxwell's equations.CRAS Compte‌ Rendu Académie des Sciences2025HAL DOI back‌ to text
21 articleT.Théophile Chaumont-Frelet and‌ A.Alexandre Ern. Damped Energy-norm a posteriori‌ error estimates using C2-reconstructions for the fully discrete‌ wave equation with the leapfrog scheme.ESAIM:‌ Mathematical Modelling and Numerical Analysis594July‌ 2025, 1937-1972HALDOI back to text‌
22 articleT.Théophile Chaumont-Frelet and M.Martin‌ Vohralík. A quasi-interpolation operator yielding fully computable‌ error bounds.ESAIM: Mathematical Modelling and Numerical‌ AnalysisNovember 2025HALDOI back to text‌
23 articleD. A.Daniele A. Di Pietro‌, Z.Zhaonan Dong, G.Guido Kanschat‌, P.Pierre Matalon and A.Andreas Rupp‌. Homogeneous multigrid for hybrid discretizations: application to‌ HHO methods.Numerical Methods for Partial Differential‌ Equations415August 2025HAL DOI
24‌ articleZ.Zhaonan Dong, A.Alexandre Ern‌ and Z.Zuodong Wang. A bound-preserving scheme‌ for the Allen-Cahn equation.Computers & Mathematics‌ with ApplicationsFebruary 2025. In press. HAL‌back to text
25 articleZ.Zhaonan Dong‌, E. H.Emmanuil H Georgoulis and P.‌ J.Philip J. Herbert. A hypocoercivity-exploiting stabilised‌ finite element method for Kolmogorov equation.SIAM‌ Journal on Numerical Analysis633May 2025‌, 1105--1127HAL DOI
26 articleJ.-P.Jean-Pierre‌ Dussault, M.Mathieu Frappier and J. C.‌Jean Charles Gilbert. Polyhedral Newton-min algorithms for‌ complementarity problems.Mathematical Programming215January 2026‌, 269-324HAL DOIback to text
27‌ articleJ.-P.Jean-Pierre Dussault, J. C.Jean‌ Charles Gilbert and B.Baptiste Plaquevent-Jourdain. On‌ the B-differential of the componentwise minimum of two‌ affine vector functions.Mathematical Programming Computation17‌1February 2025, 1-52HAL DOI back‌ to text
28 articleA.Alexandre Ern and‌ J.-L.Jean-Luc Guermond. Spectral correctness of the‌ discontinuous Galerkin approximation of the first-order form of‌ Maxwell's equations with discontinuous coefficients.SIAM Journal‌ on Numerical Analysis6322025, 661-684‌HAL DOI back to text
29 articleA.‌Alexandre Ern and J.-L.Jean-Luc Guermond. Third-order‌ sectorially A-stable alternating implicit Runge-Kutta schemes.SIAM‌ Journal on Scientific Computing4732025,‌ A1579-A1603HAL DOI back to text
30 article‌A.Alexandre Ern, J.Johnny Guzmán,‌ P.Pratyush Potu and M.Martin Vohralík.‌ Discrete Poincaré inequalities: a review on proofs, equivalent‌ formulations, and behavior of constants.IMA Journal‌ of Numerical AnalysisNovember 2025HAL DOI back‌ to text
31 articleA.Abbas Kabalan,‌ F.Fabien Casenave, F.Felipe Bordeu,‌ V.Virginie Ehrlacher and A.Alexandre Ern.‌ Elasticity-based mesh morphing technique with application to reduced-order‌ modeling.Applied Mathematical Modelling1412025,‌ 115929, 25 pp.HALDOI back to text
32 articleR.Romain‌ Mottier, A.Alexandre‌ Ern, R.Rekha‌‌ Khot and L.Laurent Guillot. Hybrid high-order‌ methods for elasto-acoustic wave‌ propagation in the time‌‌ domain.ESAIM: Mathematical Modelling and Numerical Analysis‌592025, 2685-2715‌HAL DOI back to‌‌ text
33 articleJ.Jan Nordbotten, M.‌Martin Fernø, B.‌Bernd Flemisch, A.‌‌Anthony Kovscek, K.-A.Knut-Andreas Lie, J.‌Jakub Both, O.‌Olav Møyner, T.‌‌ H.Tor Harald Sandve, E.Etienne Ahusborde‌, S.Sebastian Bauer‌, Z.Zhangxing Chen‌‌, H.Holger Class, C.Chaojie Di‌, D.Didier Ding‌, D.David Element‌‌, E.Eric Flauraud, J.Jacques Franc‌, F.Firdovsi Gasanzade‌, Y.Yousef Ghomian‌‌, M. A.Marie Ann Giddins, C.‌Christopher Green, B.‌ R.Bruno R.B. Fernandes‌‌, G.George Hadjisotiriou, G.Glenn Hammond‌, H.Hai Huang‌, D.Dickson Kachuma‌‌, M.Michel Kern, T.Timo Koch‌, P.Prasanna Krishnamurthy‌, K. O.Kjetil‌‌ Olsen Lye, D.David Landa-Marbán, M.‌Michael Nole, P.‌Paolo Orsini, N.‌‌Nicolas Ruby, P.Pablo Salinas, M.‌Mohammad Sayyafzadeh, J.‌Jakob Torben, A.‌‌Adam Turner, D.Denis Voskov, K.‌Kai Wendel and A.‌Abdallah Youssef. Benchmarking‌‌ CO2 Storage Simulations: Results from the 11th Society‌ of Petroleum Engineers Comparative‌ Solution Project.International‌‌ Journal of Greenhouse Gas Control148December 2025‌, 104519HAL DOI‌back to text back‌‌ to text back to text
34 articleM.‌Min Zhang, Z.‌Zhaonan Dong and W.‌‌Wenjing Yan. $h p$ -version discontinuous Galerkin‌ time-stepping schemes for diffusive-viscous‌ wave equation.Computers‌‌ & Mathematics with Applications200December 2025,‌ 145--166HAL DOI

International‌ peer-reviewed conferences

35 inproceedings‌‌S.Sylvie Boldo, F.François Clément,‌ D.David Hamelin,‌ M.Micaela Mayero and‌‌ P.Pierre Rousselin. Teaching Divisibility and Binomials‌ with Coq.EPTCS‌13th International Workshop on‌‌ Theorem proving components for Educational software - ThEdu‌ 2024419Nancy, France‌2025, 124 -‌‌ 139HAL DOI
36 inproceedingsK.Konstantin Brenner‌, G.Géraldine Pichot‌ and D.Daniel Zegarra‌‌ Vasquez. Multiscale and domain decomposition methods for‌ linear and nonlinear fracture‌ network flows.DD29‌‌ - 29th International Conference on Domain Decomposition Methods‌Milan, ItalyJune 2025‌HAL

Scientific book chapters‌‌

37 inbookM.Martin Vohralík and S.Soleiman‌ Yousef. A posteriori‌ error estimates and adaptivity‌‌ for locally conservative methods. Inexpensive implementation and evaluation,‌ polytopal meshes, iterative linearization‌ and algebraic solvers, and‌‌ applications to complex porous media flows.61‌Error Control, Adaptive Discretizations,‌ and Applications, Part 4‌‌Advances in Applied MechanicsElsevierNovember 2025,‌ 513-642HAL DOI back‌ to text back to‌‌ text

Doctoral dissertations and habilitation theses

38 thesis‌R.Romain Mottier.‌ Hybrid high-order methods for‌‌ the numerical simulation of‌ elasto-acoustic wave propagation.École des Ponts ParisTech‌July 2025HAL
39 thesisB.Baptiste Plaquevent-Jourdain‌. A Robust Linearization Method for Complementarity Problems.‌ A Detour Through Hyperplane Arrangements.Université de‌ Sherbrooke (Québec, Canada)July 2025HAL back to‌ text
40 thesisZ.Zuodong Wang. Efficient‌ numerical schemes for evolution equations with singularities and‌ shocks.École des Ponts ParisTechSeptember 2025‌HAL
41 thesisD.Daniel Zegarra Vasquez.‌ Efficient numerical simulation of single-phase flow in three-dimensional‌ fractured porous media.Sorbonne UniversitéMay 2025‌HAL

Reports & preprints

42 reportS.Sylvie‌ Boldo, F.François Clément, V.Vincent‌ Martin and M.Micaela Mayero. A Rocq‌ Formalization of Monomial and Graded Orders.RR-9604‌INRIADecember 2025, 17HAL back to‌ text
43 reportS.Sylvie Boldo, F.‌François Clément, V.Vincent Martin, M.‌Micaela Mayero and H.Houda Mouhcine. A‌ Rocq Formalization of Simplicial Lagrange Finite Elements.‌RR-9590INRIAJune 2025, 63HAL back‌ to text
44 miscE.Erik Burman,‌ A.Alexandre Ern and R.Romain Mottier.‌ Unfitted hybrid high-order methods stabilized by polynomial extension‌ for elliptic interface problems.October 2025HAL‌back to text
45 miscT.Théophile Chaumont-Frelet‌, M. W.Martin Werner Licht and M.‌Martin Vohralík. Computable Poincaré–Friedrichs constants for the‌ $L^{p}$ de Rham complex over convex domains‌ and domains with shellable triangulations.August 2025‌HAL
46 miscT.Théophile Chaumont-Frelet and M.‌Martin Vohralík. A stable local commuting projector‌ with local-best approximation properties in H1.November‌ 2025HAL
47 miscB.Bernardo Cockburn,‌ A.Alexandre Ern and R.Rekha Khot.‌ Improved L^2-error estimates for the wave equation discretized‌ using hybrid nonconforming methods on simplicial meshes.‌November 2025HAL back to text
48 misc‌Z.Zhaonan Dong and A.Alexandre Ern.‌ $H (𝐜𝐮𝐫𝐥)$ -reconstruction of piecewise polynomial‌ fields with application to $h p$ -a posteriori‌ nonconforming error analysis for Maxwell's equations.August‌ 2025HAL back to text
49 miscZ.‌Zhaonan Dong, E. H.Emmanuil H Georgoulis‌ and P. J.Philip J. Herbert. Asymptotic‌ numerical hypocoercivity of the space-time discontinuous Galerkin method‌ for Kolmogorov equation.May 2025HAL back‌ to text
50 miscZ.Zhaonan Dong,‌ E. H.Emmanuil H Georgoulis and N.Natalia‌ Kopteva. Adaptive robust interior penalty discontinuous Galerkin‌ methods.December 2025HAL
51 miscZ.‌Zhaonan Dong, E. H.Emmanuil H Georgoulis‌, L.Lorenzo Mascotto and Z.Zuodong Wang‌. A posteriori error analysis and adaptivity of‌ a space-time finite element method for the wave‌ equation in second order formulation.September 2025‌HAL back to text
52 miscZ.Zhaonan‌ Dong, L.Lorenzo Mascotto and Z.Zuodong‌ Wang. A priori and a posteriori error‌ estimates of a ${C}^{0}$ -in-time method for the wave equation in‌ second order formulation.‌October 2025HAL back‌‌ to text
53 miscZ.Zhaonan Dong and‌ T.Tanvi Wadhawan.‌ On the constants in‌‌ inverse trace inequalities for polynomials orthogonal to lower-order‌ subspaces.2025HAL‌
54 reportJ.-P.Jean-Pierre‌‌ Dussault, M.Mathieu Frappier and J. C.‌Jean Charles Gilbert.‌ Polyhedral Newton-min algorithms for‌‌ complementarity problems - The full report.Inria‌ Paris; Université de Sherbrooke‌ (Québec, Canada)May 2025‌‌, 57HAL
55 reportJ.-P.Jean-Pierre Dussault‌, J. C.Jean‌ Charles Gilbert and B.‌‌Baptiste Plaquevent-Jourdain. IncSignFeas.jl (isf): Julia functions for‌ the enumeration of the‌ chambers of hyperplane arrangements‌‌.Inria Paris; Université de Sherbrooke (Québec, Canada)‌September 2025HAL back‌ to text
56 report‌‌J.-P.Jean-Pierre Dussault, J. C.Jean Charles‌ Gilbert and B.Baptiste‌ Plaquevent-Jourdain. Primal and‌‌ dual approaches for the chamber enumeration of real‌ hyperplane arrangements - The‌ full report.Inria‌‌ - Paris; Université de Sherbrooke (Québec, Canada)November‌ 2025, 79HAL‌
57 reportJ.-P.Jean-Pierre‌‌ Dussault, J. C.Jean Charles Gilbert and‌ B.Baptiste Plaquevent-Jourdain.‌ Primal and dual approaches‌‌ for the chamber enumeration of real hyperplane arrangements‌.Inria - Paris;‌ Université de Sherbrooke (Québec,‌‌ Canada)November 2025, 40HAL back to‌ text
58 miscA.‌Alexandre Ern and J.-L.‌‌Jean-Luc Guermond. Convergence of explicit Runge-Kutta discontinuous‌ Galerkin approximations of the‌ first-order form of Maxwell's‌‌ equations with low regularity solutions.July 2025‌HAL back to text‌back to text
59‌‌ miscA.Alexandre Ern and J.-L.Jean-Luc Guermond‌. L2 -Stability of‌ explicit Runge-Kutta methods with‌‌ supg stabilization for transient transport problems.November‌ 2025HAL back to‌ text
60 miscA.‌‌Alexandre Ern, J.Johnny Guzmán, P.‌Pratyush Potu and M.‌Martin Vohralík. Local‌‌ L2-bounded commuting projections using discrete local problems on‌ Alfeld splits.October‌ 2025HAL back to‌‌ text back to text
61 miscJ.-L.Jean-Luc‌ Guermond and Z.Zuodong‌ Wang. Mass conservative‌‌ limiting and applications to the approximation of the‌ steady-state radiation transport equations‌.January 2025,‌‌ 113531HAL DOI back to text
62 misc‌P.Pierre Jolivet,‌ M.Michel Kern,‌‌ F.Frédéric Nataf, G.Géraldine Pichot and‌ D.Daniel Zegarra Vasquez‌. Domain decomposition preconditioners‌‌ for efficient parallel simulations of single-phase flow in‌ three-dimensional fractured porous media‌ with a very large‌‌ number of fractures.September 2025HAL back‌ to text
63 misc‌A.Abbas Kabalan,‌‌ F.Fabien Casenave, F.Felipe Bordeu,‌ V.Virginie Ehrlacher and‌ A.Alexandre Ern.‌‌ Model-order reduction with optimal morphings for poorly reducible‌ problems with geometric variability‌.December 2025HAL‌‌back to text
64 miscM.Michel Kern‌, G.Géraldine Pichot‌ and D.Daniel Zegarra‌‌ Vasquez. Mathematical and numerical analysis of the‌ mixed formulation of single‌ phase flow in three-dimensional‌‌ fractured porous media.‌October 2025HAL back to text
65 misc‌M.Michel Kern, G.Géraldine Pichot and‌ D.Daniel Zegarra Vasquez. Performance of algebraic‌ preconditioners for large-scale simulations of single-phase flow in‌ three-dimensional fractured porous media.April 2025HAL‌back to text
66 miscR.Romain Mottier‌, A.Alexandre Ern and L.Laurent Guillot‌. Elasto-acoustic wave propagation in geophysical media using‌ hybrid high-order methods on general meshes.October‌ 2025HAL back to text
67 miscJ.‌Jikun Zhao, W.Wenhao Zhu, Z.‌Zhaonan Dong and S.Shipeng Mao. The‌ pressure-robust virtual element methods with reconstruction for the‌ Stokes equation.September 2025HAL

Educational activities‌

68 unpublishedJ. C.Jean Charles Gilbert.‌ Fragments d'Optimisation Différentiable - Théories et Algorithmes.‌January 2026, MasterFranceHAL

12.3 Cited‌ publications

69 inproceedingsP.Pierre Jolivet, F.‌Frédéric Hecht, F.Frédéric Nataf and C.‌Christophe Prud'homme. Scalable 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 Machinery2013DOI back‌ to text
70 articleP.Pierre Jolivet,‌ J. E.Jose E. Roman and S.Stefano‌ Zampini. KSPHPDDM and PCHPDDM: Extending PETSc with‌ advanced Krylov methods and robust multilevel overlapping Schwarz‌ preconditioners.Computers & Mathematics with Applications84‌2021, 277-295DOIback to text

SERENA - 2025

SERENA - 2025

2025Activity report﻿﻿﻿‌Project-TeamSERENA

Keywords​‌﻿﻿

Computer Science and Digital​​﻿﻿ Science

Other Research​​﻿﻿ Topics and Application Domains​​​‌

1 Team members, visitors,​​﻿﻿ external collaborators

Research Scientists​​​‌

Post-Doctoral Fellows

PhD​‌﻿﻿ Students

Technical Staff

Administrative﻿​﻿﻿ Assistant

Visiting Scientists​​﻿﻿

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 Optimization,​‌﻿﻿ complementarity, nonsmooth analysis, discrete​​﻿﻿ geometry

3.5​​​‌ Reliability by a posteriori﻿​﻿﻿ error control

3.6﻿‌​‌ 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

7 Latest software developments,﻿​﻿﻿ platforms, open data

7.1​‌﻿﻿ Latest software developments

7.1.1​​﻿﻿ DiSk++

7.1.2﻿​﻿﻿ APS-MG

7.1.3​​​‌ FEMLAB

7.1.4​​​‌ rocq-num-analysis

7.1.5​​﻿﻿ MODFRAC

7.1.6​​​‌ nef-flow-fpm

7.1.7﻿﻿﻿‌ demonstrator-nef-hpddm

7.1.8﻿‌​‌ ParaCirce

7.1.9​‌﻿﻿ Hnm4lcp

7.1.10 ISF

7.1.11 ISF_jl﻿‌​‌

8 New results﻿﻿﻿‌

8.1 Research axis 1:﻿‌​‌ Advanced numerical discretizations and﻿​​﻿ solvers

Development and analysis​​​‌ of HHO methods

Time-stepping methods​​​‌

Invariant-domain preserving​​﻿﻿ discretizations

Complementarity problems

Discrete geometry﻿‌​‌

8.2 Research axis​​​‌ 2: A posteriori error﻿​﻿﻿ control, adaptivity, and safe​‌﻿﻿ and correct programming

A posteriori error estimates​‌﻿﻿ for the wave equation​​﻿﻿

Book​​﻿﻿ chapter on a posteriori​​​‌ error estimates

hp-a​​​‌ posteriori error estimates for﻿﻿﻿‌ nonconforming methods

Rocq formalizations﻿﻿﻿‌

8.3﻿﻿﻿‌ Research axis 3: Applications﻿‌​‌ to environment and energy​​​‌

Flow simulations in fracture﻿​﻿﻿ networks

Flow through fractured​​﻿﻿ porous media

Multiphase flow

Wave propagation﻿​​﻿ in geophysical media

Data assimilation

8.4 Research axis 4:﻿​​﻿ PDE and numerical analysis​​​‌ foundations

Stable local​​​‌ commuting projectors

Quasi-interpolation with﻿​﻿﻿ computable error bounds

Maxwell's equations

Sign-changing elliptic PDEs

Model-order﻿​​﻿ reduction with geometric variability​​​‌

Inviscid total variation and​​﻿﻿ Bingham minimizers

A hypocoercivity-exploiting stabilised﻿​﻿﻿ finite element method for​‌﻿﻿ Kolmogorov equation

9 Bilateral contracts and​​​‌ grants with industry

9.1﻿​﻿﻿ Bilateral contracts with industry​‌﻿﻿

10 Partnerships and cooperations﻿​﻿﻿

10.1 International initiatives

10.1.1​‌﻿﻿ Participation in other International​​﻿﻿ Programs

RANPDEs

AdaptPMod

10.2﻿​​﻿ International research visitors

10.2.1​​​‌ Visits of international scientists﻿﻿﻿‌

Other international visits to﻿‌​‌ the team

Jean-Luc Guermond﻿​​﻿

Buyang Li﻿​​﻿

Bernardo Cockburn

2025Activity report‌Project-TeamSERENA

Keywords‌

Computer Science and Digital Science

Other Research Topics and Application Domains‌

1 Team members, visitors, external collaborators

Research Scientists‌

PhD‌ Students

Administrative Assistant

Visiting Scientists

3 Research program‌

3.2 Advanced‌‌ numerical discretization methods

3.3 Iterative‌‌ linearization, domain decomposition, and multigrid solvers

3.4 Optimization,‌ complementarity, nonsmooth analysis, discrete geometry

3.5‌ Reliability by a posteriori error control

3.6‌‌ 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

7 Latest software developments, platforms, open data

7.1‌ Latest software developments

7.1.1 DiSk++

7.1.2 APS-MG

7.1.3‌ FEMLAB

7.1.4‌ rocq-num-analysis

7.1.5 MODFRAC

7.1.6‌ nef-flow-fpm

7.1.7‌ demonstrator-nef-hpddm

7.1.8‌‌ ParaCirce

7.1.9‌ Hnm4lcp

7.1.11 ISF_jl‌‌

8 New results‌

8.1 Research axis 1:‌‌ Advanced numerical discretizations and solvers

Development and analysis‌ of HHO methods

Time-stepping methods‌

Invariant-domain preserving discretizations

Discrete geometry‌‌

8.2 Research axis‌ 2: A posteriori error control, adaptivity, and safe‌ and correct programming

A posteriori error estimates‌ for the wave equation

Book chapter on a posteriori‌ error estimates

$h p$ -a‌ posteriori error estimates for‌ nonconforming methods

Rocq formalizations‌

8.3‌ Research axis 3: Applications‌‌ to environment and energy‌

Flow simulations in fracture networks

Flow through fractured porous media

Wave propagation in geophysical media

8.4 Research axis 4: PDE and numerical analysis‌ foundations

Stable local‌ commuting projectors

Quasi-interpolation with computable error bounds

Model-order reduction with geometric variability‌

Inviscid total variation and Bingham minimizers

A hypocoercivity-exploiting stabilised finite element method for‌ Kolmogorov equation

9 Bilateral contracts and‌ grants with industry

9.1 Bilateral contracts with industry‌

10 Partnerships and cooperations

10.1.1‌ Participation in other International Programs

10.2 International research visitors

10.2.1‌ Visits of international scientists‌

Other international visits to‌‌ the team

Jean-Luc Guermond

Buyang Li

11.1 Promoting scientific activities

11.1.1‌ Scientific events: organisation

General‌ chair, scientific chair

Member of the organizing committees‌

11.1.2 Scientific‌ events: selection

Member of the conference program committees‌

Member of the editorial boards‌

Reviewer - reviewing activities‌

11.1.5 Leadership‌ within the scientific community

11.2 Teaching - Supervision -‌ Juries - Educational and‌ pedagogical outreach

11.2.1 Supervision‌‌

11.2.3 Educational and‌‌ pedagogical outreach

11.3 Popularization

11.3.1 Participation in Live‌ events

12‌ Scientific production

12.1 Major publications

12.2 Publications of‌ the year