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.
Traditionally, we have worked in the context of finite element, finite
volume, mixed finite element, and discontinuous Galerkin methods. Novel
classes of schemes enable the use of general polygonal and polyhedral meshes with nonmatching interfaces, and we develop them in
response to a high demand from our industrial partners (namely , , and
). In the lowest-order case, our focus is to design
discrete element methods for solid mechanics. The novelty is to
devise these methods to treat dynamic elastoplasticity as well as
quasi-static and dynamic crack propagation. We also develop
structure-preserving methods for the Navier–Stokes equations,
i.e., methods that mimic algebraically at the discrete level
fundamental properties of the
underlying PDEs, such as conservation principles and preservation of
invariants. In the higher-order case, we actively
contribute to the development of hybrid high-order
methods. We contribute to
the numerical analysis in nonlinear cases (obstacle problem, Signorini
conditions), we apply these methods to challenging problems from
solid mechanics involving large deformations and plasticity, and
we develop a comprehensive software implementing them.
We believe that these methods belong to the future generation of numerical
methods for industrial simulations; as a concrete example, the
implementation of these methods in an industrial software of
has been completed in 2019 in the framework of the PhD thesis of Nicolas
Pignet.

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. Increasing complexity of algorithms for modern scientific
computing makes it a major challenge to implement them in the traditional
imperative languages popular in the community. As an alternative, the
computer science community provides theoretically sound tools for safe
and correct programming. We explore here the use of these tools to
design generic solutions for the implementation of the class of scientific
computing software that we deal with. Our focus ranges from high-level
programming via functional programming with through safe and
easy parallelism via skeleton parallel programming with to
proofs of correctness of numerical algorithms and programs via mechanical proofs with .

Via applications with our industrial and environmental partners , , , , and .

Alexandre Ern brought to completion his eight-year long project with Jean-Luc Guermond on a comprehensive book in 3 volumes on Finite Elements (83 chapters, 500 exercises, 1300 pages).

Alexandre Ern published a book, co-authored with M. Cicuttin and N. Pignet, in the collection SpringerBriefs in Mathematics on Hybrid High-Order Methods.

Many new results of the project in the .

In

, we propose a novel hybrid high-order method (HHO) to approximate singularly perturbed fourth-order PDEs on domains with a possibly curved boundary. The two key ideas in devising the method are the use of a Nitsche-type boundary penalty technique to weakly enforce the boundary conditions and a scaling of the weighting parameter in the stabilization operator that compares the singular perturbation parameter to the square of the local mesh size. With these ideas in hand, we derive stability and optimal error estimates over the whole range of values for the singular perturbation parameter, including the zero value for which a second-order elliptic problem is recovered. Numerical experiments illustrate the theoretical analysis, cf. Figure

.

In , we extend the results that we have previously established in the guaranteed), locally efficient, polynomial-degree-robust, and inexpensive. This in particular allows to certify the error committed in a numerical approximation of the simplified Maxwell problem, as well as to predict its spatial distribution, as illustrated in Figure .

In , we investigate the computational performance of hybrid high-order methods applied to flow simulations in extremely large discrete fracture networks (over one million of fractures). We study the choice of basis functions, the trade-off between increasing the polynomial order and refining the mesh, and how to take advantage of polygonal cells to reduce the number of degrees of freedom. An example of gain obtained with a polygonal discretization over a traditional triangular discretization is presented in Figure .

See Section Team members, visitors, external collaborators.

Zhaonan Dong, Alexandre Ern, and Martin Vohralík (also with Jan Papež from the Czech Academy of Sciences in Prague) co-organized this year's edition of the . The event was held in a hybrid format and hosted at Inria Paris on September 10–11, 2021.

Michel Kern organized (with Brahim Amaziane, Université de Pau) the Workshop (Advances in the SImulation of reactive flow and TRAnsport in porous Media), December 8–9, 2021, Inria Paris.

Michel Kern is a co-organizer of the minisymposium "Mathematical and numerical methods for multi-scale multi-physics, nonlinear coupled processes" at the conference.

Guillaume Enchéry and Martin Vohralík have organized a 1-day workshop /.

Alexandre Ern is a member of the organizing committee of the European Finite Element Fair.

Géraldine Pichot was a member of the Scientific Committee of the , June 21–24, 2021, Politecnico di Milano, Italy.

François Clément served as reviewer for , and for .

Alexandre Ern is a member of the editorial boards of , , , , and .

Martin Vohralík is a member of the editorial boards of and .

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

Alexandre Ern and Martin Vohralík have reviewed a dozen papers for various major journals of the field.

Michel Kern was a reviewer for SIAM Journal on Scientific Computing, Transport in Porous Media, Computer Methods in Applieds Mechanics and Engineering, Applicable Analysis and Mathematics and Computers in Simulation.

Géraldine Pichot was a reviewer for the SIAM/ASA Journal on Uncertainty Quantification.

Alexandre Ern gave one of the Plenary Talks at (virtual congress, Vienna, July 2021) and the Initial Lecture of the Australia-wide online seminar series on computational and numerical maths (February 2021).

Géraldine Pichot gave a keynote speech (50 minutes presentation) at the 2020+2021 edition of the bi-annual congress of the Italian Society of Applied and Industrial Mathematics conference in Parma (August 30 to September 3, 2021).

Martin Vohralík was an invited speaker at , online/Beyrouth, Liban, June 2021, , online/Milano, Italy, June 2021, and , September 2021, Bielefeld, Germany.

Alexandre Ern joined the Administration Board of and its Executive Committee as Vice-President in charge of relations with the industry.

Michel Kern is President of the C3I ("Certificat de Compétences en Calcul Intensif") Committee and a meber of the Scientific Board of Orap.

Martin Vohralík is a member of the steering committee of .

Martin Vohralík is a member of the scientific board of the – joint strategic partnership laboratory.

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

François Clément is a member of the Comité local d'hygiène, de sécurité et des conditions de travail of the Inria Research Center of Paris.

Michel Kern is chair of the Comission de Développement Technologique of the Inria Paris Center.

Alexandre Ern was a reviewer for the HDR defense of Aline Lefebvre-Lepot (IPP, December 2021) and a reviewer for the PhD thesis of Michele Giuliano Carlino (University of Bordeaux, December 2021). He was also the chair of the committee for the HDR of Olga Mula (Dauphine University, December 2021).

Michel Kern was an axaminer for the HDR of Laila Amir (Université de Marrakech, Morocco, April 2021), and for the PhD of Pierre Jacquet (Université de Pau, February 2021).

Géraldine Pichot was an examiner for the PhD thesis of Miao Yu (University of Paris, December 2021). She was also a member of the jury of the DT1 position: Ingénieur expérimentation et développement (h/f), Inria, September–October 2021. She also served as a member of the evaluation committee: permanent researcher position in computational geoscience, University of Bergen, February 2021.

Martin Vohralík served as referee and committee member for the PhD thesis defense of Manuela Bastidas Olivares (Hasselt University, Belgium, March 2021). He also served as referee and committee member for the PhD thesis defense of Stefan Schimanko (Vienna University of Technology, Austria, June 2021). He further served as a committee member for the PhD thesis defense of Manu Jayadharan (University of Pittsburgh, USA, July 2021). He also served as referee and committee member for the PhD thesis defense of André Harnist (Université de Montpellier, October 2021). He further served as the chair of the committee for the PhD thesis of Daria Koliesnikova (Aix Marseille Université, July 2021). He finally served as referee and committee member for the HDR sefense of Stella Krell (Université Côte d’Azur, June 2021).

Géraldine Pichot is a member of the editorial committee of Interstices.

Géraldine Pichot, Conference at the RJMI (Rendez-vous des Jeunes Mathématiciennes et Informaticiennes), November 3, 2021.