In 2022, the authors laid the foundations of a new paradigm for invariant-domain time-stepping applied to hyperbolic problems using high-order Runge–Kutta methods. The key result achieved this year is the extension to implicit-explicit (IMEX) time-stepping and the application to the compressible Navier–Stokes equations, as described in 26. The decisive step-forward is the satisfaction of physical bounds on the density and energy while allowing for a high-order discretization in space and in time. An example of application to the compressible Navier–Stokes equations at Reynolds $Re=1000$ is displayed in Figure 1. This is a very challenging problem owing to the interactions between shocks and walls and the development of multiscale vortical structures. Moreover, in 49, 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.

Further progress has been accomplished in the development and analysis of hybrid high-order (HHO) methods. Three topics were investigated. First, $C^0$-HHO methods for the biharmonic problem in 22 leading to a competitive method in terms of error vs. computational effort with respect to estabished methods such as the $C^0$-interior penalty discontinuous Galerkin method. Second, within the framework of the PhD Thesis of Morgane Steins (38,defended this year), HHO methods for the wave equation using a leapfrong scheme for time discretization were studied. The contributions include a convergence analysis 50 and a time-explicit marching scheme 35. Finally, HHO methods were used to study surface tension effects between two immiscible Stokes fluids within the PhD Thesis of Stefano Piccardo (37, defended this year) 34.

In 20, we present a local construction of $𝐇\left(\mathrm{curl}\right)$-conforming piecewise polynomials satisfying a prescribed curl constraint. We start from a piecewise polynomial not contained in the $𝐇\left(\mathrm{curl}\right)$ space but satisfying a suitable orthogonality property. The procedure employs minimizations in vertex patches and the outcome is, up to a generic constant independent of the underlying polynomial degree, as accurate as the best-approximations over the entire local versions of $𝐇\left(\mathrm{curl}\right)$. This allows to design guaranteed, fully computable, constant-free, and polynomial-degree-robust a posteriori error estimates of the Prager–Synge type for Nédélec's finite element approximations of the curl–curl problem. A divergence-free decomposition of a divergence-free $𝐇\left(\mathrm{div}\right)$-conforming piecewise polynomial, relying on over-constrained minimizations in Raviart–Thomas' spaces, is the key ingredient. Numerical results confirm the theoretical developments, see Figure 2 for an illustration.

In 29, we consider spline/isogeometric analysis discretizations of the Poisson model problem, focusing on high polynomial degrees and strong hierarchical refinements. We derive a posteriori error estimates by equilibrated fluxes, i.e., vector-valued mapped piecewise polynomials lying in the $𝐇\left(\mathrm{div}\right)$ space which appropriately approximate the desired divergence constraint. Our estimates are constant-free in the leading term, locally efficient, and robust with respect to the polynomial degree. They are also robust with respect to the number of hanging nodes arising in adaptive mesh refinement employing hierarchical B-splines. Two partitions of unity are designed, one with larger supports corresponding to the mapped splines, and one with small supports corresponding to mapped piecewise multilinear finite element hat basis functions. The equilibration is only performed on the small supports, avoiding the higher computational price of equilibration on the large supports or even the solution of a global system. Thus, the derived estimates are also as inexpensive as possible. An abstract framework for such a setting is developed, whose application to a specific situation only requests a verification of a few clearly identified assumptions. Numerical experiments illustrate the theoretical developments and even indicate, though not rigorougsly proved, robustness with respect to the smoothness of the splines.

In 28, we consider nonsmooth partial differential equations associated with a minimization of an energy functional. We adaptively regularize the nonsmooth nonlinearity so as to be able to apply the usual Newton linearization, which is not always possible otherwise. We apply the finite element method as a discretization. We focus on the choice of the regularization parameter and adjust it on the basis of an a posteriori error estimate for the difference of energies of the exact and approximate solutions. We prove guaranteed upper bounds for the energy difference, identify the individual error components, and design an adaptive algorithm with both adaptive regularization and adaptive mesh refinement. Effeciency and robustness of the estimates with respect to the magnitude of the nonlinearity is addressed in 52. Numerical results confirm the theoretical developments, see Figure 3 for an illustration.

See the "News of the Year" about software Skwer (Section 7.1.4).

In 36, we describe the formal definition and proof in Coq of product $\sigma$-algebras, product measures and their uniqueness, the construction of iterated integrals, up to Tonelli's theorem. We also advertise the Lebesgue induction principle provided by an original inductive type for nonnegative measurable functions.

See also the "News of the Year" about software coq-num-analysis (Section 7.1.5).

We have experimented with the domain decomposition preconditioner HPDDM, developped by the Alpines team  55, 56 to solve the linear system obtained from nef-flow-fpm. The improvement over the more classical preconditioners used until then (mainly algebraic multigrid) are significant as can be seen on Figure 4.

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 example, solving the linear system of size $1.41\times {10}^8$ for a network containing 378k fractures takes in parallel, with gmres preconditioned with HPDDM, only 2 minutes and 40 iterations with 4096 MPI processes.

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, 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 Alessandra Marelli's internship, we were able to simulate the transport in a network of fractures. The main challenge was the correct handling of the coupling conditions at the fracture intersections. An example is shown in Figure 5. The method is implemented in nef-transport-fpm.

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.

As part of the internship of Clément Maradei, we studied a model for the wave equations that includes both a diffusive (first order derivative in time) and a so-called "viscous" term (first order time derivative of the Laplacian). The model has been proposed to represent frequency-dependent attenuation. Thanks to finite element simulations (using FreeFeem++), we were able to compare the respective contributions of the two terms, and use a scaling analysis to better understand the influence of the two parameters. The results have been presented at the 15th FreeFem Days.

Within the PhD Thesis of Romain Mottier, 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.

Our work on data assimilation was pursued this year by addressing the heat equation. Our first contribution is on the theoretical side and concerns a Carleman estimate 17. The second 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) 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.

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. 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 first with constant coefficients 27 and then in the more challenging case of discontinuous coefficients 48.

Diffusive transport in media with discontinuous properties is a challenging problem that arises in many applications. In 39, wefocus 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 piece-wise 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 40.

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 33, we introduce an efficient algorithm to guarantee inf-sup stability for saddle-point problems with parameter-dependent constraints. In 54, we pursued a different, and complementary, approach, where the constraints are taken into account by a nonlinaer Nitsche's method, thereby allowing one to use a primal formulation. Finally, within the PhD Thesis of Abbas Kabalan, we are investigating shape variability within the context of reduced-order models.

In 23, we established optimal decay rates on the best-approximation errors using vector-valued finite elements (of Nédélec or Raviart–Thomas type) for fields with low regularity but having an integrable curl or divergence.

In 25, we derived $hp$-optimal error estimates for dG methods for the biharmonic problem with homogeneous essential boundary conditions, which removed the $1.5$ suboptimal rate in term of $p$ in the classical error analysis of dG methods. The main ingredient in the analysis is the construction of a global $H^2$ piecewise polynomial approximants with $hp$-optimal approximation properties over the meshes. 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 47. The main novelty in the analysis are novel $hp$-optimal approximation properties of the special projector introduced in [Cockburn, Dong, Guzman, SINUM, 2008].

These works were performed in collaboration with L. Mascotto.