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. SERENA has taken over the
project-team POMDAPI2 which ended on May 31, 2015. It has been given an
authorization to become a joint project-team between INRIA and ENPC at the
Committee of Projects, September 1st, 2016, and was created as project-team on April 10, 2017.

Within our project, we start from the conception and analysis of models
based on partial differential equations (PDEs). Already at the PDE
level, we address the question of coupling of different models;
examples are that of simultaneous fluid flow in a discrete network of
two-dimensional fractures and in the surrounding three-dimensional
porous medium, or that of 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. At the modeling stage, we also develop model-order reduction
techniques, such as the use of reduced basis techniques or proper generalized
decompositions, to tackle evolutive problems, in particular in the nonlinear
case, and we are also interested in developing reduced-order methods
for variational inequalities such as those encountered in solid
mechanics with contact and possibly also friction.

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

We next concentrate an intensive effort on the development and analysis of
efficient solvers for the systems of nonlinear algebraic equations that
result from the above discretizations. We have in the past developed Newton–Krylov solvers like the adaptive inexact Newton method, and we place
a particular emphasis on parallelization achieved via the domain
decomposition method. Here we traditionally specialize in Robin
transmission conditions, where an optimized choice of the parameter has
already shown speed-ups in orders of magnitude in terms of the number of
domain decomposition iterations in model cases. We concentrate in the SERENA
project on adaptation of these algorithms to the above novel discretization
schemes, on the optimization of the free Robin parameter for challenging
situations, and also on the use of the Ventcell transmission conditions.
Another feature is the use of such algorithms in time-dependent problems in
space-time domain decomposition that we have recently pioneered. This
allows the use of different time steps in different parts of the
computational domain and turns out to be particularly useful in porous media
applications, where the amount of diffusion (permeability) varies abruptly,
so that the evolution speed varies significantly from one part of the
computational domain to another. Our new theme here are Newton–multigrid solvers, where the geometric multigrid solver is tailored to the specific problem under consideration and to the specific
numerical method, with problem- and discretization-dependent restriction,
prolongation, and smoothing. Using patchwise smoothing, we have in particular recently developed a first multigrid method whose behavior is both in theory and in practice insensitive of (robust with respect to) the approximation polynomial degree. With patchwise techniques, we also achieve mass balance at each iteration step, a highly demanded feature in most of the target applications.
The solver itself is then adaptively steered at each execution step by
an a posteriori error estimate (adaptive stepsize, adaptive smoothing).

The fourth part of our theoretical efforts goes towards guaranteeing 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, from the discretization
scheme, from the nonlinear (Newton) solver, and from the linear algebraic
(Krylov, 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 also theoretically address the question of
convergence of the new fully adaptive algorithms. We identify theoretical conditions so that the error diminishes at each adaptive loop iteration by a contraction factor and we in particular derive a guaranteed error reduction factor in model cases. We have also proved a numerical optimality of the derived algorithms in model cases in the sense that, up to a generic constant, the smallest possible computational effort to achieve the given accuracy is needed.

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.

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

The ERC GATIPOR project has been finished by the concluding international workshop. We have organized the 2022 CEA–EdF–Inria summer school on Certification of errors in numerical simulations.

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.

Comparison of direct versus iterative solvers on fractured porous media of increasing complexity, the largest networks contains 23k fractures.

Add possible call to the linear direct solver MUMPS: https://mumps-solver.org/index.php

We simulate flow and transport in a network of fractures embedded in an heterogeneous porous medium. The mesh of the fractured network is generated with the software MODFRAC and constraints the volume mesh generator, GHS3D, developed by the Inria GAMMA project-team.

The coupled flow problem between the fractures and the rock matrix is solved with the mixed hybrid finite element method (Raviart-Thomas of lowest order). The method is implemented in the software nef-flow-fpm and has been validated with the established benchmark test case from 49, see Figure 1.

A particularly challenging issue is the solution of the linear system resulting from the discretization: the 3D volume mesh leads to a large number of unknowns, while the fracture network makes the problem very ill conditioned (condition numbers up to nef-flow-fpm flow in a fractured porous medium containing more than 23k fractures and meshed with 16 millions of tetrahedra (see Table 1).

Another goal is to simulate the transport by advection of an inert tracer. The first contribution was to consider a porous medium without the fractures. The transport is described by the conservation of mass and gives rise to an equation with partial derivatives of the first order in which the velocity, computed with the software nef-flow-fpm, is heterogeneous. The discretization in space is performed with a cell-centered finite volume scheme. The discretization in time can be either explicit or implicit. The method is implemented in nef-transport-fpm.

In 31, we have generalized Theorem 2.3 in 51 to the case of rectangular parallelepipeds. Then we have proposed a new initialization stage of the CEM algorithm that makes it possible to quickly jump to a domain size close to the one needed for the CEM algorithm to work. These domain size estimates are based on fitting functions. Examples of fitting functions are given for the Matérn family of covariances. These functions are inspired by our numerical simulations and by the theoretical work from 51. The parameters estimation of the fitting functions is done numerically. Several numerical tests are performed to show the efficiency of the proposed algorithms, for both isotropic and anisotropic Matérn covariances.

Examples of 2D fields with isotropic and anisotropic Gaussian covariances are shown on Figure 3.

In 50, we provide detailed proofs for the main results
in measure theory and Lebesgue integration theory, to help for their
formalization in an interactive theorem prover such as Coq.
The document gathers material from several textbooks, but also provides some
original views on known concepts such as set systems (e.g. Coq formalizations presented
in 15 and 33 (see below).
Version 3 (December 2022, added 450 statements and 130 pages) brings more
abstractions, for instance for set systems and numeric functions over set
systems (e.g. measures), for properties satisfied almost everywhere, and for
simple functions.
It also generalizes the chapter on

In 15, we present the Coq formalization of
Coq theorems.
These results are a first milestone towards the formalization of

In 33, we describe the formal definition and proof in
Coq of product Lebesgue induction principle provided by an
original inductive type for nonnegative measurable functions.

In 35, we describe the formalization of Bochner integration, a generalization of Lebesgue integration to functions taking their values in a Banach space, up to the dominated convergence theorem. Our contributions include an original formalization of simple functions, Bochner integrability defined by a dependent type, and the proof of the integrability of measurable functions under a weak separability hypothesis.

In 43, the authors started the development of a novel paradigm laying the foundations of invariant-domain time-stepping for hyperbolic problems using high-order Runge–Kutta methods. Further developments in progress include the extension to implicit-explicit (IMEX) time-stepping and the application to the compressible Navier–Stokes equations.

In 28 we develop a space-time mortar mixed finite element method for parabolic problems. The domain is decomposed into a union of subdomains discretized with non-matching spatial grids and asynchronous time steps. The method is based on a space-time variational formulation that couples mixed finite elements in space with discontinuous Galerkin in time. Continuity of flux (mass conservation) across space-time interfaces is imposed via a coarse-scale space-time mortar variable that approximates the primary variable. Uniqueness, existence, and stability, as well as a priori error estimates for the spatial and temporal errors, are established. A space-time non-overlapping domain decomposition method is developed that reduces the global problem to a space-time coarse-scale mortar interface problem. Each interface iteration involves solving in parallel space-time subdomain problems. The spectral properties of the interface operator and the convergence of the interface iteration are analyzed.

Numerical experiments are provided that illustrate the theoretical results and the flexibility of the method for modeling problems with features that are localized in space and time, see Figure 4. This work was in collaboration with I. Yotov and M. Jayadharan.

In 41, the authors proposed a convergence theory for high-order multiscale FEMs independent of the roughness of the diffusion tensor for Poisson problems. We aim to provide a priori and a posteriori error analysis for high-order discontinuous Galerkin multiscale methods for flows problems on heterogeneous domains.

This work was in collaboration with R. Maier and M. Hauck.

In 38, the authors present a new residual-type energy-norm a posteriori error analysis for interior penalty discontinuous Galerkin (dG) methods for linear elliptic problems. The new error bounds are also applicable to dG methods on meshes consisting of elements with very general polygonal/polyhedral shapes. The case of simplicial and/or box-type elements is included in the analysis as a special case. In particular, for the upper bounds, arbitrary number of very small faces are allowed on each polygonal/polyhedral element, as long as certain mild shape regularity assumptions are satisfied. As a corollary, the present analysis generalizes known a posteriori error bounds for dG methods, allowing in particular for meshes with arbitrary number of irregular hanging nodes per element. The proof hinges on a new conforming recovery strategy in conjunction with a Helmholtz decomposition formula. The resulting a posteriori error bound involves jumps on the tangential derivatives along elemental faces.

Numerical experiments presented in Figure 5 highligts the practical value of the derived a posteriori error bounds as error estimators. This work was performed in collaboration with A. Cangiani and E. H. Georgoulis.

In collaboration with K. Mitra, we have in 47 studied the Richards equation. This equation is recognized as principal model for flows in underground porous media. It describes more precisely the flow of water and air through subsoil and serves as a gateway for multiphase flows through porous media. This nonlinear advection–reaction–diffusion equation exhibits both parabolic–hyperbolic and parabolic–elliptic degeneracies, which make its numerical approximation and mathematical analysis very challenging.

In this study, we provide guaranteed, fully computable, and locally space–time efficient a posteriori error bounds for numerical approximations of the doubly-degenerate Richards equation. For showing global reliability, a nonlocal-in-time error estimate is derived individually for the time-integrated

Numerical tests are conducted for nondegenerate and degenerate cases having exact solutions, as well as for a realistic/benchmark case. Excellent results are obtained even in this last setting, see Figure 6. The error elementiwse distribution (right) varies here for more than 5 orders of magnitude in the spatial domain at each given time, but it is still correctly predicted by the estimators (middle). The overall space-time estimator only overestimates the error by the factor of 1.05.

Prof. Johnny Guzmán (Brown University, USA) visited the team for two weeks in October 2022. He gave a seminar at the Scientific computing, modeling, and numerical analysis seminar “Rencontres Inria-LJLL en calcul scientifique” (common to Sorbonne University and Inria) and interacted with Alexandre Ern and Martin Vohralík on two topics: (i) local- and global-best approximations in H(div); (ii) localized stability estimates for convective problems without diffusion.

Prof. Jean-Luc Guermond (Texas A&M University) visited the team for about two months in 2022. He mainly interacted with Alexandre Ern on invariant-domain time-stepping methods, but he also interacted with Zhaonan Dong on local interpolation estimates in H(curl) and H(div) 23. He also gave a seminar at the working group of the team.

EMC2 project (or on CORDIS)

Zhaonan Dong, Alexandre Ern, André Harnist, Ani Miraçi, Ari Rappaport, and Martin Vohralík (also with Jan Papež from the Czech Academy of Sciences in Prague) co-organized the 2022 workshop concluding the ERC consolidator grant GATIPOR.

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

Michel Kern was a co-organizer of the minisymposium “Mathematical and numerical methods for multi-scale multi-physics, nonlinear coupled processes” at the Interpore 2022 conference.

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

Martin Vohralík co-organized, with Mickaël Abbas (EDF R&D), Ludovic Chamoin (ENS Paris-Saclay), and Erell Jamelot, François Madiot, and Pascal Omnes (all CEA Saclay), the CEA-EdF-Inria Summer School on Certification of errors in numerical simulations, held from 27 June to 01 July 2022 at EDF Lab, Paris-Saclay.

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

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

Zhaonan Dong was a reviewer for the SIAM Journal on Numerical Analysis, SIAM Journal on Scientific Computing, and Mathematics of Computation.

Alexandre Ern and Martin Vohralík reviewed numerous papers for the leading journals of the field.

Michel Kern reviewed papers for Mathematics and Computers in Similation, Hydrology and Earth System Science, and Computer Methods in Applied Mathematics.

Géraldine Pichot served as reviewer for Engineering Geology.

Alexandre Ern gave a plenary lecture at the POEMS 2022 conference in Milano (December 2022). He also gave an invited talk at the One-World Numerical Analysis seminar.

Michel Kern gave a seminar in the PIMENT team at the Université de la Réunion (December 2022).

Martin Vohralík gave an invited talk at the 34th seminar on Numerical fluid mechanics (January 2022), a keynote talk at Equadiff 15, Brno, Czech Republic (July 2022), and an invited talk at the Singular Days, Nice, France (November 2022).

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