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 EDF, CEA, and
IFP Energies Nouvelles). 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 EDF
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 OCaml through safe and
easy parallelism via skeleton parallel programming with Sklml to
proofs of correctness of numerical algorithms and programs via mechanical proofs with Coq.

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

Many new results of the ERC GATIPOR project in the ERC GATIPOR Gallery.

Python library for the simulation of reactive transport in porous media.

The library couples a transport module with a geochemistry module. The transport module is provided by the ComPASS code (developed by BRGM and Inria) and the chemistry module is developed as part of this project.

In 11, we completed the study in 1 by giving the details of a global in time domain-decomposition (DD) method for a problem modelling two-phase flow with discontinuous capillary pressure. We proved the existence of a solution for the local problem (a nonlinea degenerate parabolic problem with nonlinear Robin boundary conditions). The proof is by convergence of a finite volume scheme, and uses new compactness results by [Andreianov, Cancès, and Moussa, J. Funct. Anal. 273 (2017)]. We also studied in detail the convergence of the Optiomized Schwarz Waveform Relaxation (OSWR) algorithm.

In

50, we extend the applicability of the popular interior-penalty discontinuous Galerkin (dG) method discretizing advection-diffusion-reaction problems to meshes comprising extremely general, essentially arbitrarily-shaped elemens. In particular, our analysis allows for curved element shapes, without the use of non-linear elemental maps. The feasibility of the method relies on the definition of a suitable choice of the discontinuity penalization, which turns out to be explicitly dependent on the particular element shape, but essentially independent on small shape variations. This is achieved upon proving extensions of classical trace inverse estimate to arbitrary element shapes. A further new

–

-type inverse estimate on essentially arbitrary element shapes enables the proof of inf-sup stability of the method in a streamline diffusion-like norm. These inverse estimates may be of independent interest. A priori error bounds for the resulting method are given under very mild structural assumptions restricting the magnitude of the local curvature of element boundaries. Moreover, we have also developed a GPU library for the massive parallel implemetation of the proposed DG methods, see

57.

In

26, we introduce a residual-based a posteriori error estimator for a novel

-version interior penalty discontinuous Galerkin method for the biharmonic problem in two and three dimensions. We prove that the error estimate provides an upper bound and a local lower bound on the error, and that the lower bound is robust to the local mesh size but not the local polynomial degree. The suboptimality in terms of the polynomial degree is fully explicit and grows at most algebraically. Our analysis does not require the existence of a

-conforming piecewise polynomial space and is instead based on an elliptic reconstruction of the discrete solution to the

space and a generalised Helmholtz decomposition of the error. This is a

-version error estimator for the biharmonic problem in two and three dimensions. The practical behavior of the estimator is investigated through numerical examples in two and three dimensions.

In

36, we continue our developments started in

35. We in particular develop an iterative algebraic solver of the multigrid type, whose distinctive feature is that it is steered by an a posteriori estimator of the algebraic error. We work in the context of a second-order elliptic diffusion problem discretized by conforming finite elements of arbitrary polynomial degree

. In contrast to an established practice, our solver employs no pre-smoothing and solely one post-smoothing step, carried over via the overlapping Schwarz (block-Jacobi) method. Moreover, it features an optimal choice of the step-sizes in the smoothing correction on each level by line search. This leads to a simple Pythagorean formula of the algebraic error in the next step in terms of the current error and level-wise and patch-wise error reductions. We show the two following results and their equivalence: the solver contracts the algebraic error independently of the polynomial degree

; and the estimator represents a two-sided

-robust bound on the algebraic error. We consider quasi-uniform or graded bisection simplicial meshes and prove mild dependence on the number of mesh levels for minimal

-regularity and complete independence for

-regularity. We also present a simple and effective way for the solver to adaptively choose the number of post-smoothing steps necessary at each individual level, yielding a yet improved error reduction. Numerical tests confirm

-robustness and show the benefits of the adaptive number of smoothing steps. An illustration of the performance of the developed solver is illustrated in Figure

4(red boxes), in comparision with some established algorithms.

University College London, Great Britain, Texas A&M University, USA, UPC (Spain), Institute of Mathematics, Czech Academy of Sciences, Charles University, Prague, Ecole Nationale d'Ingénieurs de Tunis

GiS: scientific collaboration network between ten public institutions from the Paris (Ile-de-France) region, focused on natural resources and environment. The project-team SERENA is a member.