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 coq-num-analysis

• Name:
  Numerical analysis Coq library
• Keywords:
  Formal methods, Coq, Numerical analysis, Finite element modelling
• Scientific Description:
  These Coq developments are based on the Coquelicot library for real analysis. Version 1.0 includes the formalization and proof of: (1) the Lax-Milgram theorem, including results from linear algebra, geometry, functional analysis and Hilbert spaces, (2) the Lebesgue integral, including large parts of the measure theory,the building of the Lebesgue measure on real numbers, integration of nonnegative measurable functions with the Beppo Levi (monotone convergence) theorem, Fatou's lemma, the Tonelli theorem, and the Bochner integral with the dominated convergence theorem.
• Functional Description:
  Formal developments and proofs in Coq of numerical analysis problems. The current long-term goal is to formally prove parts of a C++ library implementing the Finite Element Method.
• News of the Year:

  The formalization in Coq of simplicial Lagrange finite elements is complete. This include the formalizations of the definitions and main properties of monomials, their representation using multi-indices, Lagrange polynomials, the vector space of polynomials of given maximum degree (about 6 kloc). This also includes algebraic complements on the formalization of the definitions and main properties of operators on finite families of any type, the specific cases of abelian monoids (sum), vector spaces (linear combination), and affine spaces (affine combination, barycenter, affine mapping), sub-algebraic structures, and basics of finite dimension linear algebra (about 31 kloc). A new version (2.0) of the opam package will be available soon, and a paper will follow.

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

• URL:
  https://­lipn.­univ-paris13.­fr/­coq-num-analysis/
• Publications:
  hal-01344090, hal-01391578, hal-03105815, hal-03471095, hal-03516749, hal-03889276, hal-04713897, tel-04884651
• Contact:
  Sylvie Boldo
• Participants:
  Sylvie Boldo, François Clement, Micaela Mayero, Vincent Martin, Stéphane Aubry, Florian Faissole, Houda Mouhcine, Louise Leclerc

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:
  Addition of the ability to update the mesh with a size map. Better management of 1D and 2D constraints imposed on the surface of the cube.
• Publications:
  hal-03480570, hal-02102811
• 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 data per subdomain for HPDDM solver (Neumann matrices, local sizes and indices, local second member), add an option to assemble the linear system with the boundary conditions (BCs) and save the necessary data so that the Dirichlet BCs are removed in parallel in PETSc. Simulations with the demonstrator for PETSc/HPDDM, the largest networks contains 696k fractures (and 243M unknowns).
• URL:
  https://­gitlab.­inria.­fr/­nef/­nef-flow-fpm
• Contact:
  Geraldine Pichot
• Participants:
  Geraldine Pichot, Daniel Zegarra Vasquez, Michel Kern, Raphael Zanella

7.1.7 nef-transport-fpm

• Keywords:
  3D, Porous media, Incompressible flows, Transport model
• Scientific Description:

  The discretization in space is performed with a cell-centered finite volume scheme. The discretization in time can be either explicit or implicit.

  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-transport-fpm is a Matlab code for simulating transport by advection diffusion in porous-fractured media.
• News of the Year:
  Add transport in a fractured porous medium with the proper handling of the coupling conditions between the fractures (2D) and the rock matrix (3D) and the coupling conditions at the fractures intersections (1D). Validation of the implementation by comparison with the Field test case described in Inga Berre, et al., 2021 (https://doi.org/10.1016/j.advwatres.2020.103759).
• Contact:
  Geraldine Pichot
• Participants:
  Geraldine Pichot, Michel Kern, Daniel Zegarra Vasquez, Alessandra Marelli, Dania Khiralla

7.1.8 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 text 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.

• URL:
  https://­gitlab.­inria.­fr/­nef/­demonstrator-nef-hpddm
• Contact:
  Geraldine Pichot
• Participants:
  Geraldine Pichot, Raphael Zanella

7.1.9 ParaCirce

• Name:
  Parallel Circulant Embedding
• Keywords:
  2D, 3D, Hydrogeology, Gaussian random fields, MPI
• Scientific Description:
  ParaCirce implements the algorithm proposed by [C. R. Dietrich and G. N. Newsam. A fast and exact method for multidimensional gaussian stochastic simulations. Water Resources Research, 29(8):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:
  - Simple precision fields generation support - HDF5 format fields export
• News of the Year:
  - Export fields in HDF5 format - Add simple precision support - WIP: Support of multidimensional communicators
• URL:
  https://­gitlab.­inria.­fr/­slegrand/­paracirce
• Publication:
  hal-03190252
• Contact:
  Geraldine Pichot
• Participants:
  Geraldine Pichot, Simon Legrand

7.1.10 Pruners

• Name:
  Pruners
• Keyword:
  Parameter studies
• Functional Description:
  Pruners is a language aimed at automating parameter studies. It allows the specification of parameter combinations, and make them available via environment variables. Those can then be used by any specified command as input parameters.
• Release Contributions:
  - Bug fix: Json arrays support - Bug Fix: Curly brackets aroud variables are properly removes when configuring
• News of the Year:
  - Bug fix: Json arrays support - Bug Fix: Curly brackets aroud variables are properly removes when configuring
• URL:
  https://­team.­inria.­fr/­serena/­en/­research/­software/­pruners/
• Contact:
  Thierry Martinez

7.1.11 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

7.1.12 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

7.1.13 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

Invariant-domain time-stepping for compressible flows

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

In 2022, Alexandre Ern and Jean-Luc Guermond laid the foundations of a new paradigm for invariant-domain time-stepping applied to hyperbolic problems using high-order Runge–Kutta methods. The main application in mind are the compressible Navier–Stokes equations. The main contribution this year in this direction is the devising of sectorially A-stable alternating implicit Runge–Kutta schemes of third-order 55. This has been a long-standing barrier for decades, where only second-order accuracy was available together with some form of A-stability.

Moreover, in 25, we considered a scalar conservation law with a stiff source term having multiple equilibrium points. For this quite challenging situation, we proposed a scheme that can be asymptotic-preserving.

Analysis of HHO methods

Participants: Zhaonan Dong, Alexandre Ern, Rekha Khot.

Hybrid high-order (HHO) methods play still an important role in our research endeavors. One contribution 20 is the devising and analysis of a $C^0$-HHO method for the biharmonic problem. A second one 26 is the fully discrete analysis of the HHO space discretization combined with the explicit leapfrog scheme for the acoustic wave equation in its second-order form. The challenge here was to handle the static coupling between cell and face unknowns. A third contribution 58 is the fully discrete analysis of explicit Runge–Kutta schemes combined with HHO methods for the acoustic wave equation in its first-order form. Two main steps forward are (i) a unified vision of abstract assumptions underlying the error analysis; (ii) a broad appraoch for interpolation operators, encompassing several nown operators from either the HHO or Hybridizable discontinuous Galerkin (HDG) context, thereby providing a further unifying bridge between HHO and HDG.

Stabilized primal-dual method for 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 results on challenging problems are reported in 41. Results for the cloaking using a metamaterial are reported in Figure 1; one can see the striking role of the cloak on the right panel.

      

Show image description

No cloaking vs. cloaking with metamaterial.

Show image description

No cloaking vs. cloaking with metamaterial.

Multigrid and domain decomposition solvers steered by a posteriori estimates of the algebraic error

Participants: Akram Beni Hamad, Martin Vohralík.

In 38 and 63, we consider iterative algebraic solvers for saddle-point mixed finite element discretizations of arbitrary polynomial degree $p\ge 0$ of the model Darcy flow problem. We propose 1) a posteriori error estimators of the algebraic error; and 2) novel iterative algebraic solvers. The estimators control the algebraic error from above and from below in a guaranteed and fully computable way. The solvers are then steered by these estimators. The distinctive feature of the approaches in both 38 and 63 is that they produce a locally mass conservative approximation on each iteration. This is easier in 63, where a divergence-free approximation is constructed on each iteration of a multigid or an overlapping domain decomposition approach (updating a suitable initial guess), but ingenious in 38, where a nonoverlapping domain decomposition framework is considered. A line search is employed in both cases to determine the optimal step size, leading to a Pythagoras formula for the algebraic error decrease in each iteration. A multilevel stable decomposition of the velocity space is designed in 63 to prove a $p$-robust contraction of the algebraic error on each iteration and, simultaneously, the efficiency of the associated lower algebraic a posteriori estimator. Both approaches are suitable for massive parallelization. Numerical experiments illustrate the theoretical developments. Figure 2 from 38 illustrates a fast convergence for a model problem with repeated diffusion contrast of size ${10}^7$, tight computable upper and lower bounds on the algebraic error, and an accurate prediction of the distribution of the algebraic error.

      

      

Show image description

Algebraic errors and estimates

Show image description

Algebraic errors and estimates

Complementarity problems

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

The contributions 22, 52 propose 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 $H\left(x\right):=min\left(F\right(x),G(x\left)\right)=0$, and takes the associated least-square function $x\mapsto \theta \left(x\right):=\frac{1}{2}{\parallel H\left(x\right)\parallel }_2^2$ as merit function, which means that the progress to the solution is measured by the imposed decrease of $\theta$. 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 $\theta$. 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 (§ 7.1.11) 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 67, this approach is applied to the nonsmooth system $H\left(x\right)=0$ 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 $\theta$ at a point (i.e., ${\theta }^{\text{'}}(x;d)⩾0$, for all $d$) is co-NP-complete. Finding a point satisfying a weaker form of stationarity (i.e., $0\in {\partial }_C\theta \left(x\right)$), however, may be feasible under suitable assumptions.

Nonsmooth analysis

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

The contributions 23, 53 propose an algorithm for computing the B-differential (roughly speaking, the set of Jacobians found at points very near the considered point) of the componentwise minimum of two affine functions, which is the map $x\in {ℝ}^n\mapsto min(Ax+a,Bx+b)\in {ℝ}^m$, where $A$, $B\in {ℝ}^{m\times n}$, $a$, $b\in {ℝ}^m$ and the minimum function “$min$” acts componentwise. The B-differential of this function is a finite set of ${ℝ}^{m\times n}$ that can have a number of elements that is exponential in $m$. Along the way, we identify several equivalent problems in linear algebra, convex analysis and discrete geometry. Several properties of the B-differential, like its symmetry, conditions for its completeness, its connectivity, bounds on its cardinality, etc, are also stated and proved. It is the hyperplane arrangement formulation that is the most appropriate for calculation. With this point of view, we improve significantly an algorithm proposed by Rada and Černy in 2018 (the speed-up can exceed 100, but this one depends much on the considered instance) and propose a dual approach based on Gordan's theorem of the alternative, matroid circuits and the introduced notion of stem vectors. The Matlab code isf (§  7.1.12) has been developed to validate the proposed algorithms.

Discrete geometry

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

The hyperplanes in 23, 53 are linear, meaning that they all contain zero. The contribution 68 extends the approach proposed in 23 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 (§ 7.1.13) has been written to assess the efficiency of the algorithms proposed to enumerate the chambers of an affine arrangement.

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 44, 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.

A posteriori error estimates for the DG-CG time-stepping method combined with the finite element method (FEM) in space, applied to the wave equation, are presented in 51. These estimates are formulated in the $L^{\infty }\left(L^2\right)$-norm, providing a constant-free upper bound that is optimal with respect to both the mesh size and the polynomial degree.

A posteriori error estimates and adaptivity for nonsmooth problems

Participants: Ari Rappaport, Martin Vohralík.

In 28, we develop guaranteed a posteriori error estimates for nonlinear problems with nonsmooth nonlinearities and design adaptive regularization algorithms that automatically apply only the amount of regularization necessary not to harm the overall precision. This makes the Newton method converge in situations where it would fail usually and enables to keep long time steps for evolutive problems. In 27, we apply this approach to the doubly-degenerate Richards equation (a central model for saturated-unsaturated flows in porous media). Figure 3 presents the instantaneous computational gains, where the “no harm of the overall precision” is presented graphically. These results were obtained in the Ph.D. thesis of Ari Rappaport 37 defended this year.

      

Show image description

A posteriori error estimates and adaptivity for nonsmooth problems

Show image description

A posteriori error estimates and adaptivity for nonsmooth problems

Coq formalizations

Participants: François Clément, Houda Mouhcine, Pierre Rousselin.

In 45, we provide very detailed pen-and-paper proofs for the construction of the Lagrange finite elements of any degree on simplicies in positive dimension. The first version of the document covers: the general definition of finite element, the construction of simplicial Lagrange finite elements on a segment including results about ${\mathbb{P}}_k^1$ Lagrange polynomials, and the construction of simplicial Lagrange finite elements in dimension $d\ge 1$ including the construction of multi-indices of given maximum length, results about multivariate polynomials such as the linear independence of monomials and the Euclidean division by a monomial, results about ${\mathbb{P}}_1^d$ Lagrange polynomials such as their view as barycentric coordinates, results about affine geometric mappings such as the transformation of $l$-faces of dimension $l\le d$, results about Lagrange nodes and Lagrange linear forms of ${\mathbb{P}}_k^d$, and the proofs of unisolvence of ${\mathbb{P}}_k^d$, and of face unisolvence.

These detailed proofs served as a guideline for the formalization in Coq of the simplicial Lagrange finite elements during the Ph.D. thesis of Houda Mouhcine 36 defended this year. See also the "News of the Year" about software coq-num-analysis (Section 7.1.4).

In the context of the Inria Challenge LiberAbaci, and during the half-delegation of Pierre Rousselin in the project-team SERENA, two papers were presented at the 13th international workshop on Theorem proving components for Educational software (ThEdu'24). In 34, we present worksheets in Coq for students to learn about divisibility and binomial coefficients. Technical and pedagogical choices are presented, as well as numerous exercises. In 35, we report on a 3-year teaching experiment conducted at Sorbonne Paris Nord University. The subject was an introductory course in formal proofs with the Coq proof assistant given to first-year students in a double major mathematics and computer science curriculum. Practical session worksheets are presented, and the evolution of the course and improvements in both technical and pedagogical aspects are discussed.

Flow through fractured and fractured porous media

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

We have continued the work with the domain decomposition preconditioner HPDDM, developped by the Alpines team 69, 70 to solve the linear system obtained from nef-flow-fpm(§ 7.1.6). We demonstrate that HPDDM outperforms other more classical preconditioners used until then (mainly algebraic multigrid) as can be seen on Figure 4.

Show image description

Performances of the domain decomposition preconditioner HPDDM

Thanks to the GMRES solver and the HPDDM preconditioner, we are now able to solve flow problem in large scale fractured porous media that are out of reach with direct solvers like MUMPS Cholesky or with GMRES preconditioned by multigrid like BoomerAMG. As 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 simulations are run on the TGCC (GENCI / CEA).

Another goal is to simulate the transport by advection of an inert tracer. The transport is described by the conservation of mass and gives rise to an equation with partial derivatives of the first order in which the velocity, computed with the software nef-flow-fpm(§ 7.1.6), is heterogeneous. The discretization in space is performed with a cell-centered finite volume scheme. The discretization in time can be either explicit or implicit.

As part of Tall Ouba 's internship, we are now able to simulate the transport in a fractured porous medium. The main challenge was the correct handling of the coupling conditions between the rock and the fractures . An example is shown in Figure 5. The method is implemented in nef-transport-fpm(§ 7.1.7).

Show image description

Concentration in the porous medium and in the discrete fracture network with 52 fractures.

Multiphase flow

Participants: Michel Kern, Brahim Amaziane, Dorian Nicou.

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

Show image description

Evolution of the gas saturation

Show image description

Evolution of the gas saturation

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 teams presented results for all three test cases that were in line with the majority of other groups, see Figure 7. A synthesis article is in preparation.

Show image description

CO2 molar fraction ,SPE CSP tcase B

Show image description

CO2 molar fraction ,SPE CSP tcase B

Wave propagation in geophysical media

Participants: Alexandre Ern, Romain Mottier.

Within the Ph.D. thesis of Romain Mottier (to be submitted soon), 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 8.

Show image description

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) 40. 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.

A stable local commuting projector and optimal $hp$ approximation estimates in $𝐇\left(\mathrm{curl}\right)$

Participants: Théophile Chaumont-Frelet, Martin Vohralík.

In 18, we design an operator from the infinite-dimensional Sobolev space $𝐇\left(\mathrm{curl}\right)$ to its finite-dimensional subspace formed by the Nédélec piecewise polynomials on a tetrahedral mesh that has the following properties: 1) it is defined over the entire $𝐇\left(\mathrm{curl}\right)$, including boundary conditions imposed on a part of the boundary; 2) it is defined locally in a neighborhood of each mesh element; 3) it is based on simple piecewise polynomial projections; 4) it is stable in the ${𝐋}^2$-norm, up to data oscillation; 5) it has optimal (local-best) approximation properties; 6) it satisfies the commuting property with its sibling operator on $𝐇\left(\mathrm{div}\right)$; 7) it is a projector, i.e., it leaves intact objects that are already in the Nédélec piecewise polynomial space. This operator can be used in various parts of numerical analysis related to the $𝐇\left(\mathrm{curl}\right)$ space. We in particular employ it here to establish the two following results: i) equivalence of global-best, tangential-trace- and curl-constrained, and local-best, unconstrained approximations in $𝐇\left(\mathrm{curl}\right)$ including data oscillation terms; and ii) fully $h$- and $p$- (mesh-size- and polynomial-degree-) optimal approximation bounds valid under the minimal Sobolev regularity only requested elementwise.

Maxwell's equations

Participants: Alexandre Ern, Jean-Luc Guermond.

In 43, we established the asymptotic optimality of the edge finite element approximation of the time-harmonic Maxwell's equations. This fundamental result, which is the counterpart of a known result concering the Helmholtz equation and conforming finite elements, was still lacking in the litterature. As a further development, we also established in 42 a similar result for the discontinuous Galerkin 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 24. The analysis is quite challenging, and relies upon a duality argument and a deflated inf-sup condition. As a further extension, we showed in 54 that the same results holds for a continuus finite element discretization combined with edge stabilization on special meshes (for instance, based on Alfeld splits of given simplicial meshes).

Solutions to 1D advection-diffusion problems with discontinuous coefficients

Participants: Géraldine Pichot.

Diffusive transport in media with discontinuous properties is a challenging problem that arises in many applications. In 15, we focus on one-dimensional discontinuous media with generalized permeable boundary conditions at the discontinuity interface. The paper presents novel analytical expressions from the method of images to simulate diffusive processes, such as mass or thermal transport. The analytical expressions are used to formulate a generalization of the existing Skew Brownian Motion, HYMLA and Uffink's method, here named as GSBM, GHYMLA, and GUM respectively, to handle generic interface conditions. The algorithms rely upon the random walk method and are tested by simulating transport in a bimaterial and in a multilayered medium with piecewise constant properties. The results indicate that the GUM algorithm provides the best performance in terms of accuracy and computational cost. The methods proposed can be applied for simulation of a wide range of differential problems, like heat transport problem 16.

Model-order reduction

Participants: Alexandre Ern, Abbas Kabalan.

One important topic has been the development of reduced-order methods to handle variational inequalities such as those encountered when studying contact problems (with friction) in computational mechanics. In 32, we introduce a new approach in the reduced-order modeling context, where the constraints are taken into account by a nonlinaer Nitsche's method, thereby allowing one to use a primal formulation. Another important topic, explored within the Ph.D. thesis of Abbas Kabalan , is the construction of morphings between a reference shape and a collection of target shapes. The main achievement is to devise an efficient version of the approach based on an offline/online decomposition. The approach can also be used to learn scalar outputs on variable geometries. The methodology and associated numerical results are reported in 62.

$hp$-optimal error estimates for nonconforming methods

Participants: Zhaonan Dong, Alexandre Ern.

In 48, we derived $hp$-a posteriori error estimates for the HHO method applied to a variable-diffusion problem. The main breakthrough is the use of local Helmholtz decompositions combined with a partition-of-unity technique to bound the nonconformity error. Novel insights were also gained by leveraging the conservative property of HHO methods and exploiting equilibrated fluxes in the derivation of the error upper bound.

Moreover, recently, we derived $hp$-optimal error estimates for the upwind dG method when approximating solutions to first-order hyperbolic problems with constant convection fields in the $L_2$ and DG norms in 21. The main novelty in the analysis are novel $hp$-optimal approximation properties of the special projector introduced in [Cockburn, Dong, Guzmán, SINUM, 2008]. These works were performed in collaboration with L. Mascotto.

A hypocoercivity-exploiting stabilised finite element method for Kolmogorov equation

Participants: Zhaonan Dong.

In 50, we propose a new stabilised FEM for the Kolmogorov equation. The latter serves as a basic model problem for large classes of kinetic-type equations and, crucially, is characterised by degenerate diffusion. The stabilisation is constructed so that the resulting method admits a numerical hypocoercivity property, analogous to the corresponding property of the PDE problem. More specifically, the stabilisation is constructed so that spectral gap is possible in the resulting “stronger-than-energy” stabilisation norm, despite the degenerate nature of the diffusion in Kolmogorov, thereby the method has a provably robust behaviour as the “time” variable goes to infinity, see Figure 9 for an illustration. We consider both a spatially discrete version of the stabilised finite element method and a fully discrete version, with the time discretisation realised by discontinuous Galerkin timestepping. Both stability and a priori error bounds are proven in all cases. Numerical experiments verify the theoretical findings. These works were performed in collaboration with P. Herbert and E. H. Georgoulis.

Show image description

Kolmogorov decay to equilibrium

Show image description

Kolmogorov decay to equilibrium

10.1.1 Visits of international scientists

10.1.2 Visits to international teams

11.1.1 Scientific events: organisation

• Alexandre Ern co-organized (with Paul Vigneaux and Nicolas Seguin) a one-week workshop at Nouan-le-Fuzelier (November 2024) in the framework of the CNRS Thematic Network on Earth and Energies.
• Zhaonan Dong , Michel Kern , Géraldine Pichot and Nicolas Pignet organized the workshop “Polytopal Element Methods in Mathematics and Engineering” (POEMS2024). It took place on December 3rd – 6th, 2024 at the Inria Paris center, and gathered close to 100 participants from over 10 countries, from academia and industry.
• Martin Vohralík (with Guillaume Enchéry, IFP Energies Nouvelles) organized the regular 1-day workshop Journée contrat cadre IFP Energies Nouvelles – Inria.

11.1.2 Journal

11.1.3 Invited talks

Alexandre Ern was invited to give the 2024 Von Mises Lecture in Berlin in July 2024. He was also invited to give a Keynote Lecture (the only one in Numerical Analysis) at the 9th Pacific Rim Conference in Mathematics held in Darwin in June 2024. Martin Vohralík was an invited speaker of the “Mathematical Models and Numerical Methods for Multiphysics Systems” conference in Pittsburgh in May 2024 and was invited to deliver the SIAM Geosciences Webinar in September 2024.

11.1.4 Leadership within the scientific community

• Michel Kern is a member of the Scientific Board of ORAP, Organisation Associative du Parallélisme.
• Michel Kern serves at the steering committee of GDR HydroGEMM.
• 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.

11.1.5 Scientific expertise

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

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

11.1.6 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 13ème Forum Entreprises et Mathématiques (FEM2024)
• Géraldine Pichot is a member of the Conseil du Département Mathématiques Appliquées Polytech Lyon.

11.2.1 Teaching

• 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, Models and numerical methods for subsurface flow, 30h, M2, Université Paris Saclay, France.
• Master (M1): Michel Kern, Advanced numerical analysis, 30h, M1, Institut Galilée, Université Paris-Nord, France.
• Master (M1): Martin Vohralík, Advanced finite elements, 21h, M1, ENSTA (Ecole nationale supérieure de techniques avancées), Paris, France.
• Master (M2): Martin Vohralík, A posteriori error estimates via equilibrated fluxes, 36h, M2, Charles University, Prague, Czech Republic.

11.2.2 Supervision

• Ph.D. defended: Houda Mouhcine , Formal Proofs in Applied Mathematics: A Coq Formalization of Simplicial Lagrange Finite Elements, Decembre 9th, 2024, supervised by Sylvie Boldo (Toccata), François Clément, Micaela Mayero (LIPN), 36
• Ph.D. defended: Ari Rappaport, A posteriori error estimates and adaptivity in numerical approximation of PDEs: regularization, linearization, discretization, and floating point precision, March 22, 2024, supervised by François Févotte (TriScale innov) and Martin Vohralík, 37
• 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: Abbas Kabalan , Model order reduction for nonparametrized geometrical variability, started October 2022, supervised by Virginie Ehrlacher and Alexandre Ern
• Ph.D. in progress: Clément Maradei , Robust a posteriori error control and adaptivity with inexact solvers, started October 2023, supervised by Zhaonan Dong and Martin Vohralík
• Ph.D. in progress: Romain Mottier , Hybrid nonconforing methods for elasto-acoustic wave propagation and unfitted meshes, started November 2021, defense expected April 2025, supervised by Alexandre Ern and Laurent Guillot (CEA)
• Ph.D. in progress: Baptiste Plaquevent-Jourdain , Robust methods for solving complementarity problems, started January 2022, Sorbonne University and Université de Sherbrooke (Canada), supervised by Jean-Pierre Dussault (Université de Sherbrooke) and Jean Charles Gilbert
• Ph.D. in progress: Zuodong Wang , Finite element methods for hyperbolic and degenerate parabolic problems, started October 2022, supervised by Alexandre Ern and Zhaonan Dong
• Ph.D. in progress: Daniel Zegarra Vasquez , High-performance simulation of single-phase flows in a fractured porous medium, started October 2021, Géraldine Pichot , Michel Kern , and Martin Vohralík .
• Internship: Dorian Nicou , Master “Analyse, Modélisation et Simultion”, Université Paris-Saclay, April–September 2024, supervised by Michel Kern
• Internship: Tall Ouba, Simulation of advective transport in fractured porous media, June–July 2024, supervised by Géraldine Pichot and Michel Kern

11.2.3 Juries

• Alexandre Ern was member of the Ph.D. defense committees for Joyce Ganthous (University of Pau), Loic Balazi (IP Paris), Benoit Cossart (University of Bordeaux) and Robin Roussel (Sorbonne University)
• Martin Vohralík was member of the Ph.D. defense committee for Miranda Boutilier (Université Côte d'Azur)

11.3.1 Participation in Live events

Michel Kern visited Lycée Lucie Aubrac (Courbevoie, 92) as part of the "1 scientifique, 1 classe: Chiche !" program. Michel Kern , Géraldine Pichot , and Martin Vohralík participated at the “accueil des stagiaires de 3ème” at Inria Paris.

International journals

• 14 articleE.Etienne Ahusborde, B.Brahim Amaziane, S.Stephan de Hoop, M.Mustapha El Ossmani, E.Eric Flauraud, F.François Hamon, M.Michel Kern, A.Adrien Socié, D.Danyang Su, K. U.K. Ulrich Mayer, M.Michal Tóth and D.Denis Voskov. A benchmark study on reactive two-phase flow in porous media : Part II -results and discussion.Computational Geosciences283February 2024, 395-412HALDOIback to text
• 15 articleE.Elisa Baioni, A.Antoine Lejay, G.Géraldine Pichot and G. M.Giovanni Michele Porta. Modeling diffusion in discontinuous media under generalized interface conditions: theory and algorithms.SIAM Journal on Scientific Computing464July 2024, A2202-A2223HALDOIback to text
• 16 articleE.Elisa Baioni, A.Antoine Lejay, G.Géraldine Pichot and G. M.Giovanni Michele Porta. Random walk modeling of conductive heat transport in discontinuous media.Transport in Porous Media2024. In press. HALDOIback to text
• 17 articleA.Annalisa Buffa, O.Ondine Chanon, D.Denise Grappein, R.Rafael Vázquez and M.Martin Vohralík. An equilibrated flux a posteriori error estimator for defeaturing problems.SIAM Journal on Numerical Analysis626November 2024, 2439-2458HALDOI
• 18 articleT.Théophile Chaumont-Frelet and M.Martin Vohralík. A stable local commuting projector and optimal hp approximation estimates in H(curl).Numerische Mathematik15662024, 2293-2342HALDOIback to text
• 19 articleT.Théophile Chaumont-Frelet and M.Martin Vohralík. Constrained and unconstrained stable discrete minimizations for p-robust local reconstructions in vertex patches in the de Rham complex.Foundations of Computational MathematicsNovember 2024HALDOI
• 20 articleZ.Zhaonan Dong and A.Alexandre Ern. $C^0$-hybrid high-order methods for biharmonic problems.IMA Journal of Numerical Analysis441February 2024, 24–57HALDOIback to text
• 21 articleZ.Zhaonan Dong and L.Lorenzo Mascotto. $hp$-optimal convergence of the original DG method for linear hyperbolic problems on special simplicial meshes.IMA Journal of Numerical AnalysisAugust 2024. In press. HALDOIback to text
• 22 articleJ.-P.Jean-Pierre Dussault, M.Mathieu Frappier and J. C.Jean Charles Gilbert. Polyhedral Newton-min algorithms for complementarity problems.Mathematical Programming(in revision)November 2024HALback to text
• 23 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 ComputationAugust 2024. In press. HALDOIback to textback to textback to text
• 24 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 Analysis2024. In press. HALback to text
• 25 articleA.Alexandre Ern, J.-L.Jean-Luc Guermond and Z.Zuodong Wang. Asymptotic and invariant-domain preserving schemes for scalar conservation equations with stiff source terms and multiple equilibrium points.Journal of Scientific Computing10032024, Paper No. 83, 30ppHALback to text
• 26 articleA.Alexandre Ern and M.Morgane Steins. Convergence analysis for the wave equation discretized with hybrid methods in space (HHO, HDG and WG) and the leapfrog scheme in time.Journal of Scientific Computing10112024, Paper 7, 28 ppHALDOIback to text
• 27 articleF.François Févotte, A.Ari Rappaport and M.Martin Vohralík. Adaptive regularization for the Richards equation.Computational Geosciences286September 2024, 1371-1388HALDOIback to text
• 28 articleF.François Févotte, A.Ari Rappaport and M.Martin Vohralík. Adaptive regularization, discretization, and linearization for nonsmooth problems based on primal-dual gap estimators.Computer Methods in Applied Mechanics and Engineering418BJanuary 2024, 116558HALDOIback to text
• 29 articleG.Gregor Gantner and M.Martin Vohralík. Inexpensive polynomial-degree-robust equilibrated flux a posteriori estimates for isogeometric analysis.Mathematical Models and Methods in Applied Sciences343January 2024, 477-522In press. HALDOI
• 30 articleS.Stephan de Hoop, D.Denis Voskov, E.Etienne Ahusborde, B.Brahim Amaziane and M.Michel Kern. A benchmark study on reactive two-phase flow in porous media: Part I -model description.Computational GeosciencesJanuary 2024HALDOIback to text
• 31 articleK.Koondanibha Mitra and M.Martin Vohralík. A posteriori error estimates for the Richards equation.Mathematics of Computation93347February 2024, 1053-1096HALDOI
• 32 articleI.Idrissa Niakh, G.Guillaume Drouet, V.Virginie Ehrlacher and A.Alexandre Ern. A reduced basis method for frictional contact problems formulated with Nitsche's method.SMAI Journal of Computational Mathematics10June 2024, 29-54HALback to text

International peer-reviewed conferences

• 33 inproceedingsB.Bernardo Cockburn, D. A.Daniele Antonio Di Pietro and A.Alexandre Ern. Bridging the Hybrid High-Order and Hybridizable Discontinuous Galerkin Methods: Summary.Frontiers of Science Awards for Math/TCIS/PhysICBS 2024 - International Congress on Basic ScienceBeiing, ChinaInternational PressJuly 2024HAL

Conferences without proceedings

• 34 inproceedingsS.Sylvie Boldo, F.François Clément, D.David Hamelin, M.Micaela Mayero and P.Pierre Rousselin. Teaching Divisibility and Binomials with Coq.13th International Workshop on Theorem proving components for Educational software - ThEdu 2024Nancy, FranceJuly 2024HALback to text
• 35 inproceedingsM.Marie Kerjean, M.Micaela Mayero and P.Pierre Rousselin. Maths with Coq in L1, a pedagogical experiment.ThEdu 2024 - 13th International Workshop on Theorem proving components for Educational softwareNancy, FranceJuly 2024HALback to text

Doctoral dissertations and habilitation theses

• 36 thesisH.Houda Mouhcine. Formal Proofs in Applied Mathematics : A Coq Formalization of Simplicial Lagrange Finite Elements.Université Paris-SaclayDecember 2024HALback to textback to text
• 37 thesisA.Ari Rappaport. A posteriori error estimates and adaptivity in numerical approximation of PDEs : regularization, linearization, discretization, and floating point precision.Sorbonne UniversitéMarch 2024HALback to textback to text

Reports & preprints

• 38 miscM.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.December 2024HALback to textback to textback to textback to text
• 39 reportS.Sylvie Boldo, F.François Clément, D.David Hamelin, M.Micaela Mayero and P.Pierre Rousselin. Teaching Divisibility and Binomials with Coq.RR-9547InriaApril 2024, 13HAL
• 40 miscE.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.July 2024HALDOIback to text
• 41 miscE.Erik Burman, A.Alexandre Ern and J.Janosch Preuss. A stabilized hybridized Nitsche method for sign-changing elliptic PDEs.December 2024HALback to text
• 42 miscT.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.May 2024HALDOIback to text
• 43 miscT.Théophile Chaumont-Frelet and A.Alexandre Ern. Asymptotic optimality of the edge finite element approximation of the time-harmonic Maxwell's equations.May 2024HALback to text
• 44 miscT.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.March 2024HALback to text
• 45 reportF.François Clément and V.Vincent Martin. Finite element method. Detailed proofs to be formalized in Coq.RR-9557Inria ParisSeptember 2024, 172HALback to text
• 46 miscL. F.Leszek F. Demkowicz and M.Martin Vohralík. $p$-robust equivalence of global continuous constrained and local discontinuous unconstrained approximation, a $p$-stable local commuting projector, and optimal elementwise $hp$ approximation estimates in $H\left(\mathrm{div}\right)$.March 2024HAL
• 47 miscD. 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.April 2024HALDOI
• 48 miscZ.Zhaonan Dong and A.Alexandre Ern. $hp$-error analysis of mixed-order hybrid high-order methods for elliptic problems on simplicial meshes.October 2024HALback to text
• 49 miscZ.Zhaonan Dong, A.Alexandre Ern and Z.Zuodong Wang. A bound-preserving scheme for the Allen-Cahn equation.February 2025HAL
• 50 miscZ.Zhaonan Dong, E. H.Emmanuil H Georgoulis and P. J.Philip J. Herbert. A hypocoercivity-exploiting stabilised finite element method for Kolmogorov equation.December 2024HALback to text
• 51 miscZ.Zhaonan Dong, L.Lorenzo Mascotto and Z.Zuodong Wang. A priori and a posteriori error estimates of a DG-CG method for the wave equation in second order formulation.November 2024HALback to text
• 52 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)March 2024, 57HALback to textback to text
• 53 reportJ.-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 - The full report.Inria de Paris, France & Université de Sherbrooke, CanadaAugust 2024HALback to textback to text
• 54 miscA.Alexandre Ern and J.-L.Jean-Luc Guermond. Spectral correctness of the continuous finite element stabilized approximation of the first-order form of Maxwell's equations.February 2024HALback to text
• 55 miscA.Alexandre Ern and J.-L.Jean-Luc Guermond. Third-order A-stable alternating implicit Runge-Kutta schemes.March 2024HALback to text
• 56 miscA.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.December 2024HAL
• 57 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.February 2025HAL
• 58 miscA.Alexandre Ern and R.Rekha Khot. Explicit Runge-Kutta schemes with hybrid high-order methods for the wave equation in first-order form.November 2024HALback to text
• 59 miscF.François Févotte and A.Ari Rappaport. A thread parallel implementation of the equilibrated flux in Julia.April 2024HAL
• 60 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, 113531HALDOI
• 61 miscA.André Harnist, K.Koondanibha Mitra, A.Ari Rappaport and M.Martin Vohralík. Robust augmented energy a posteriori estimates for Lipschitz and strongly monotone elliptic problems.June 2024HAL
• 62 miscA.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.2024HALDOIback to text
• 63 miscA.Ani Miraçi, J.Jan Papež, M.Martin Vohralík and I.Ivan Yotov. A-posteriori-steered $p$-robust multigrid and domain decomposition methods with optimal step-sizes for mixed finite element discretizations of elliptic problems.June 2024HALback to textback to textback to textback to text
• 64 miscR.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.February 2025HAL
• 65 miscM.Martin Vohralík. $p$-robust equivalence of global continuous and local discontinuous approximation, a $p$-stable local projector, and optimal elementwise $hp$ approximation estimates in $H^1$.February 2024HAL

Software

• 66 softwareJ.-P.Jean-Pierre Dussault and J. C.Jean Charles Gilbert. Hnm4lcp - A solver of linear complementarity problems using the hybrid Newton-min algorithm.1.0November 2024Inria - Paris; Université de Sherbrooke (Québec, Canada) lic: QPL 1.0.HALSoftware HeritageVCS