3.2.1 Core research topics

Our research activities are organized around core theoretical and methodological topics to address the above-listed modeling challenges.

Multiscale modeling. The physical models that we consider may feature three different space scales. First, the size of the computational domain is fixed by the nanostructure under consideration and the required observables. Second, the solution wavelength depends on the operating frequency and on the light velocity in the constitutive materials. Finally, the finest scale involved corresponds to the nanostructuring length. These three space scales can differ by orders of magnitude, leading to unaffordable computational costs, if the discretization scheme must resolve the nanostructure details. We thus aim at providing multiscale numerical schemes, that can embed fine scale information into a coarser mesh. Such methodologies lead to embarrassingly parallel two-level algorithms, that are especially suited for HPC environments and produce accurate numerical approximations. These multiscale schemes will in particular be designed in the framework of MHM (Multiscale Hybrid-Mixed) formulations recently introduced in collaboration with Frédéric Valentin at LNCC, Petropolis, Brazil. In the MHM framework, the inherent upscaling procedure that is at the heart of the approach, allows to incorporate more physics in the numerical schemes themselves, as the upscaling principle is used to construct physical basis functions that resolve the fine scales. At the second level, one defines a set of boundary value problems, whose solutions call for adapted versions of classical finite element or DG methods, and yield the upscaled basis functions.

High order DG methods. Designing numerical schemes that are high order accurate on general meshes, i.e., unstructured or hybrid structured/unstructured meshes, will still represent a major objective of our core research activities in ATLANTIS. We will focus on the family of Discontinuous Galerkin (DG) methods that has been extensively developed for wave propagation problems during the last 15 years. We will investigate several variants, namely nodal DGTD for time-domain problems, and HDG (Hybridized DG) for time-domain and frequency-domain problems, with the general goal of devising, analyzing and developing extensions of these methods, in order to deal with the above-mentioned physical drivers: nonlinear features, in particular in relation with generation of higher order harmonics in electromagnetic wave interaction with nonlinear materials, and nonlinear models of electronic response in metallic and semiconductor materials; multiphysic couplings such as for instance when considering PDE models relevant to thermoplasmonics, optoelectronics and nanoelectronics. There are to date very few works promoting DG-type methods for these situations. Concerning nonlinear effects, it is worth notice that DG methods have already proved to be attractive and have been extensively developed for computational fluid dynamics, and in particular for gas dynamics. The development of a high order DG method for situations involving nonlinear materials require to address several issues such as the construction approximate Riemann solvers for the evaluation of numerical traces associated with the evaluation of boundary integral terms in DG weak forms, the formulation of positivity-preserving schemes as continuous solutions may develop nonlinear discontinuities and the definition of linearization strategies in conjunction with time-integration schemes.

Error estimation and adaptivity. While standard theory ensures the convergence of the discrete approximation to the correct solution, it does not permit to quantitatively estimate the discretization error in actual applications. As a result, the selection of the discretization parameters is often carried out by the practitioners themselves, based on their experience and hence manual approaches to guarantee a sufficient accuracy level. We aim at providing reliable measurements of the discretization error and devising a more systematic and rigorous procedure to select and adapt discretization parameters — in particular the mesh size and the discretization order — through the use of a posteriori estimators. On the one hand, we propose to design a posteriori error estimators for the wave-matter interaction problems we consider in ATLANTIS, able to provide a fully reliable error estimation. On the other hand, such estimators can be employed to drive hp-adaptive algorithms, where the mesh and discretization order are iteratively improved to fit the complicated structure of the solution.

Time integration for multiscale problems. Multiscale physical problems with complex geometries or heterogeneous media are extremely challenging for conventional numerical simulations. Adaptive mesh refinement is an attractive technique for treating such problems and will be developed in our research activities in ATLANTIS. Local mesh refinement imposes a severe stability condition on explicit time integration since the allowed maximal time step size is constrained by the smallest element in the mesh. We will focus on different ways to overcome this stability condition, especially by using implicit-explicit (IMEX) methods where a time implicit scheme is used only for the refined part of the mesh, and a time explicit scheme is used for the other part. Optimizing the CPU time requires coupling IMEX methods with hybridized DG (HDG) methods and non-conforming meshes and will be one of the main objectives of our research activities in ATLANTIS.

Dealing with complex materials. Physically relevant simulations deal with increasing levels of complexity in the geometrical and/or physical characteristics of nanostructures, as well as their interaction with light. Standard simulation methods may fail to reproduce the underlying physical phenomena, therefore motivating the search for more sophisticated light-matter interaction numerical modeling strategies. A first direction consists in refining classical linear dispersion models and we shall put a special focus on deriving a complete hierarchy of models, that will encompass standard linear models to more complex and nonlinear ones (such as Kerr-type materials, non- linear quantuum hydrodynamic theory models, etc.). One approach will rely on an accurate description of the Hamiltonian dynamics with intricate kinetic and exchange correlation energies, for different modeling purposes. A second direction will be motivated by the study of 2D materials. A major concern is centered around the choice of the modeling approach between a full costly 3D modeling and the use of equivalent boundary conditions (the so-called GSTC), that could in all generality be nonlinear. Assessing these two directions will require efficient dedicated numerical algorithms that will especially be able to tackle several types of nonlinearities and scales.

Dealing with coupled models. Several of our target physical fields are multiphysics in essence and require going beyond the sole description of the electromagnetic response. In thermoplasmonics, the various phenomena (heat transfer through light concentration, bubbles formation and dynamics) call for different kinds of governing PDEs (Maxwell, conduction, fluid dynamics). Since, in addition, these phenomena can occur in significatively different space and time scales, drawing a quite complete picture of the underlying physics is a challenging task, both in terms of modeling and numerical treatment. In the nanoelectronics field, an accurate description of the electronic properties involves including quantum effects. A coupling between Maxwell’s and Schrödinger equations (again at significantly different time and space scales) is a possible relevant scenario. In the optoelectronics field, the accurate prediction of semiconductors optical properties is a major concern. A possible strategy may require to solve both the electromagnetic and the drift-diffusion equations. In all these aforementioned examples, difficulties mainly arise both from the differences in physical nature as well as in the time/space scales at which each physical phenomenon occurs. Accurately modeling/solving their coupled interactions remains a formidable challenge.

High performance computing (HPC). HPC is transversal to almost all the other research topics considered in the team, and is concerned with both numerical algorithm design and software development. We will work toward taking advantage of fine grain massively parallel processing offered by GPUs in modern exascale architectures, by revisiting the algorithmic structure of the computationally intensive numerical kernels of the high order DG-based solvers that we develop. We will mostly rely on the lastest features of the OpenMP standard for implementing these numerical algorithms in the DIOGENeS software suite, which already offers a coarse grain parallelization based on MPI. We will also investigate task-based fine grain parallelization for dealing with load balancing issues inherent to realistic wave-matter interaction problems and that are sourced by both numerical (e.g., locally adapted interpolation order, PML layers, volumic observables, etc.) and physical (e.g., multiple material models) characteristics.

3.2.2 Complementary topics

Beside the above-discussed core reserach topics, we have also identified additional topics that are important or compulsory in view of maximizing the impact in nanophotonics or nanophononics of our core activities and methodological contributions.

Inverse design. Inverse design has emerged rather recently in nanophotonics, and is currently the subject of intense research as witnessed by several reviews 55. Artificial Intelligence (AI) techniques are also increasingly investigated within this context 63. In ATLANTIS, we will extend the modeling capabilities of the DIOGENeS software suite by using statistical learning techniques for the inverse design of nanophotonic devices. When it is linked to the simulation of a realistic 3D problem making use of one of the high order DG and HDG solvers we develop, the evaluation of a figure of merit is a costly process. Since a sufficiently large input data set of candidate designs, as required by using Deep Learning (DL), is generally not available, global optimization strategies relying on Gaussian Process (GP) models are considered in the first place. This activity will be conducted in close collaboration with researchers of the ACUMES project-team. In particular, we investigate GP-based inverse design strategies that were initially developed for optimization studies in relation with fluid flow problems 43- 44 and fluid-structure interaction problems 61.

Uncertainty analysis and quantification. The automatic inverse design of nanophotonic devices enables scientists and engineers to explore a wide design space and to maximize a device performance. However, due to the large uncertainty in the nanofabrication process, one may not be able to obtain a deterministic value of the objective, and the objective may vary dramatically with respect to a small variation in uncertain parameters. Therefore, one has to take into account the uncertainty in simulations and adopt a robust design model 48. We study this topic in close collaboration with researchers of the ACUMES project-team one one hand, and researchers at TU Braunschweig in Germany.

Numerical linear algebra. Sparse linear systems routinely appear when discretizing frequency-domain wave-matter interaction PDE problems. In the past, we have considered direct methods, as well as domain decomposition preconditioning coupled with iterative algorithms to solve such linear systems 7. In the future, we would like to further enhance the efficiency of our solvers by considering state-of-the-art linear algebra techniques such as block Krylov subspace methods 34, or low-rank compression techniques 60. We will also focus on multi-incidence problems in periodic structures, that are relevant to metagrating or metasurface design. Indeed, such problems lead to the resolution of several sparse linear systems that slightly differ from one another and could benefit from dedicated solution algorithms. We will collaborate with researchers of the HIEPACS (Inria Bordeaux-Sud Ouest) project-team to develop efficient and scalable solution strategies for such questions.

5.1.1 DIOGENeS

• Name:
  DIscOntinuous GalErkin Nanoscale Solvers
• Keywords:
  High-Performance Computing, Computational electromagnetics, Discontinuous Galerkin, Computational nanophotonics
• Functional Description:
  The DIOGENeS software suite provides several tools and solvers for the numerical resolution of light-matter interactions at nanometer scales. A choice can be made between time-domain (DGTD solver) and frequency-domain (HDGFD solver) depending on the problem. The available sources, material laws and observables are very well suited to nano-optics and nano-plasmonics (interaction with metals). A parallel implementation allows to consider large problems on dedicated cluster-like architectures.
• URL:
  https://­diogenes.­inria.­fr/
• Authors:
  Stéphane Lanteri, Nikolai Schmitt, Alexis Gobé, Jonathan Viquerat, Guillaume Leroy
• Contact:
  Stéphane Lanteri

6.1.1 Reduced-order modeling for time-domain electromagnetics

Participants: Stéphane Lanteri [UESTC, Chengdu, China], Kun Li [UESTC, Chengdu, China], Liang Li [UESTC, Chengdu, China].

In short, reduced order models (ROMs) are simplifications of high fidelity, complex models. They capture the behavior of these source models so that one can quickly study a system's dominant effects using minimal computational resources. During the last four years, in collaboration with researchers at the University of Electronic Science and Technology of China and the Southwestern University of Finance and Economics, Chengdu in China, we have studied reduced-order modeling for time-domain electromagnetics and nanophotonics. More precisely, we have considered the applicability of the proper orthogonal decomposition (POD) technique for the system of 3D time-domain Maxwell equations, possibly coupled to a Drude dispersion model, which is employed to describe the interaction of light with nanometer scale metallic structures. A reduced subspace with a significantly smaller dimension is constructed by a set of POD basis vectors extracted offline from snapshots that are extracted from simulations with a high order DGTD method. In our first contribution, a POD-based ROM is established by projecting (Galerkin projection) the global semi-discrete DG scheme onto the low-dimensional space 51- 52. Our most recent achievement 18 introduces a non-intrusive variant of the POD-based ROM the solution of parameterized time-domain electromagnetic scattering problems. Here, the considered parameters are the electric permittivity and the temporal variable. By using the singular value decomposition (SVD) method, the principal components of the projection coefficient matrices (also referred to as the reduced coefficient matrices) of full-order solutions onto the RB subspace are extracted. A cubic spline interpolation-based (CSI) approach is proposed to approximate the dominating time- and parameter-modes of the reduced coefficient matrices without resorting to Galerkin projection. The generation of snapshot vectors, the construction of POD basis functions and the approximation of reduced coefficient matrices based on the CSI method are completed during the offline stage. The RB solutions for new time and parameter values can be rapidly recovered via outputs from the interpolation models in the online stage. In particular, the offline and online stages of the proposed RB method, termed as the POD-CSI method, are completely decoupled, which ensures the computational validity of the method. Moreover, a surrogate error model is constructed as an efficient error estimator for the POD-CSI method.

6.1.2 Dynamic metasurface control using Deep Reinforcement Learning

Participants: Stéphane Lanteri [UESTC, Chengdu, China], Kun Li [UESTC, Chengdu, China], Liang Li [UESTC, Chengdu, China], Jonathan Viquerat [CEMEF, Mines ParisTech, Sophia Antipolis, France], Ying Zhao [UESTC, Chengdu, China].

Dynamic metasurface is an emerging concept for achieving a flexible control of electromagnetic waves. Generalized sheet transition conditions (GSTCs) can be used to model the relationship between the electromagnetic response and surface susceptibility parameters characterizing a metasurface. However, when it comes to the inverse problem of designing and controlling a metasurface in a space-time varying context based on GSTCs, the dynamic synthesis of the susceptibility parameters is a difficult and non-intuitive task. In this work, we transform the inverse problem of solving dynamic susceptibility parameters into a sequence of control problems. Based on FDTD numerical simulations, a Deep Reinforcement Learning (DRL) framework using a proximal policy optimization (PPO) algorithm and a fully connected neural network is designed to control the susceptibility parameters intelligently and efficiently, promoting the further expansion of the application range of metasurface and thus helping realize more flexible and effective control of electromagnetic waves. We provide numerical results in a one-dimensional setting to show the applicability, correctness and effectiveness of the proposed approach.

6.1.3 Stability and asymptotic properties of the linearized Hydrodynamic Drude model

Participants: Serge Nicaise [Université de Valenciennes], Claire Scheid.

We go a step further toward a better understanding of the fundamental properties of the linearized hydrodynamical model studied. This model is especially relevant for small nanoplasmonic structures (below 10 nm). Using a hydrodynamical description of the electron cloud, both retardation effects and non local spatial response are taken into account. This results in a coupled PDE system for which we study the linear response. In 58, we concentrated on establishing well posedness results combined to a theoretical and numerical stability analysis. We especially proved polynomial stability and provided optimal energy decay rate. Finally, we investigated the question of numerical stability of several explicit time integration strategies combined to a Discontinuous Galerkin spatial discretization. This year, we have also extended the theoretical results of well posedness and stability to a problem of propagation in an open metallic domain with a hole (using the same model). A publication is currently in the second reviewing step after minor revisions.

6.1.4 Toward thermoplasmonics

Participants: Yves D'Angelo, Stéphane Lanteri, Thibault Laufroy, Claire Scheid.

Although losses in metal are viewed as a serious drawback in many plasmonics experiments, ther- moplasmonics is the field of physics that tries to take advantage of the latter. Indeed, the strong field enhancement obtained in nanometallic structures lead to a localized temperature increase in its vicin- ity, leading to interesting photothermal effects. Therefore, metallic nanoparticles may be used as heat sources that can be easily integrated in various environments. This is especially appealing in the field of nanomedecine and can for example be used for diagnosis purposes or nanosurgery to cite but just a few. Due to the various scales and phenomena that come into play, accurate numerical modeling is challenging. Laser illumination first excites a plasmon oscillation (reaction of the electrons of the metal) that relaxes to a thermal equilibrium and in turn excites the metal lattice (phonons). The latter is then responsible for heating the surroundings. A relevant modeling approach thus consists in describing the electron-phonon coupling through the evolution of their respective temperature. Maxwell’s equations are then coupled to a set of coupled nonlinear hyperbolic equations for the temperatures of respectively electrons, phonons and environment. The nonlinearities and the different time scales at which each thermalization occurs make the numerical approximation of these equations quite challenging. In the context of the PhD of Thibault Laufroy, which has started in October 2020, we propose to develop a suitable numerical framework for studying thermoplasmonics. As a first step, we have reviewed the models used in thermoplasmonics that are most often based on strong or weak (nonlinear) couplings of Maxwell’s equations with nonlinear equations modeling heat transfer (hyperbolic or parabolic). We also layed the foundations of the numerical approximation framework and provided a first stability study. This activity is planed to be carried out in close collaboration with physicists or chemists specialized in the field, in particular from the Fresnel Institute in Marseille (Guillaume Baffou).

6.1.5 Corner effects in nanoplasmonics

Participants: Camille Carvalho [Applied Mathematics Department, University of California Merced, USA], Patrick Ciarlet [Inria POEMS project-team and ENSTA, Paris], Claire Scheid.

In this work, we study nanoplasmonic structures with corners (typically a diedral/triangular structure); a situation that raises a lot of issues. We focus on a lossles Drude dispersion model and propose to investigate the range of validity of the amplitude limit principle. The latter predicts the asymptotic harmonic regime of a structure that is monochromatically illuminated, which makes a frequency domain approach relevant. However, in frequency domain, several well posedness problems arise due to the presence of corners (addressed in the PhD thesis of Camille Carvalho). This should impact the validity of the limit amplitude principle and has not yet been addressed in the literature in this precise setting. Here, we combine frequency-domain and time-domain viewpoints to give a numerical answer to this question in two dimensions. We show that the limit amplitude principle does not hold for whole interval of frequencies, that are explicited using the well-posedness analysis. This work has been published in 10.

6.1.6 DGTD method for modeling EM wave propagation in a gain medium

Participants: Stéphane Descombes, Stéphane Lanteri, Cédric Legrand.

In laser physics, gain or amplification is a process where the medium transfers part of its energy to an incident electromagnetic radiation, resulting in an increase in optical power. This is the basic principle of all lasers. Quantitatively, gain is a measure of the ability of a laser medium to increase optical power. Modeling optical gain requires to study the interaction of the atomic structure of the medium with the incident electromagnetic wave. Indeed, electrons and their interactions with electromagnetic fields are important in our understanding of chemistry and physics. In the classical view, the energy of an electron orbiting an atomic nucleus is larger for orbits further from the nucleus of an atom. However, quantum mechanical effects force electrons to take on discrete positions in orbitals. Thus, electrons are found in specific energy levels of an atom. In a semiclassical setting, such transitions between atomic energy levels are generally described by the so-called rate equations (RE). These rate equations model the behavior of a gain material, and they need to be solved self-consistently with the system of Maxwell equations. So far, the resulting coupled system of Maxwell-RE equations has mostly been considered in a time-domain setting using the FDTD method for which several extensions have been proposed. In the context of the PhD of Cédric Legrand, we study an alternative numerical modeling approach by relying on our know how of the DGTD method. This year, we have proposed a first variant that extends the DGTD method introduced in 45. The method has been formulated and analyzed in the three-dimensional case, and the computer implementation is underway.

6.1.7 DGTD method for semiconductor device modeling

Participants: Stéphane Lanteri, Massimiliano Montone, Claire Scheid.

In the field of semiconductor physics modeling, charge carrier transport is the starring phenomenon that needs to be predicted in order to build a mathematical model, based on higher-level quantities (e.g. electric current and voltage), that can be practically used for device simulation. Charge carrier transport is generally described by a drift-diffusion model. This yields a system of partial differential equations (PDEs) which can be solved at two levels: (1) quasistatic approximation: the external force applied to the crystal is electrostatic, and drift-diffusion equations are coupled to a Poisson equation for the electrostatic potential. The goal is to determine the spatial distribution of carrier concentrations and the electric field (deduced from the potential); (2) fullwave model: the crystal is subject to an applied electromagnetic field, and Maxwell equations are coupled with transport equations for carrier dynamics. The goal is to determine the space-time evolution of carrier concentrations and the electromagnetic field. The quasistatic approximation is rigorous when the steady state of the semiconductor has to be calculated. For example, this could be a preliminary step to the fullwave simulation of a device that is biased prior to responding to a (time-varying) electromagnetic excitation. The fullwave model is particularly relevant to electro-optics, i.e. when light-matter interaction is investigated. Indeed, such study is essential to understanding and accurately modeling the operation of photonic devices for light generation, modulation, absorption. In the context of the PhD of Massimiliano Montone we have a designed a DGTD methd for solving the the coupled system of Maxwell equations and drift-diffusion equations in the fullwave setting. The method has been formulated in the general three-dimensional case, and a computer implementation has been realized in a two-dimensional setting.

6.1.8 Advanced modeling of nanostructured CMOS imagers based on DG methods

Participants: Jérémy Grebot [STMicroelectronics, Crolles], Stéphane Lanteri [STMicroelectronics, Crolles], Denis Rideau [STMicroelectronics, Crolles], Claire Scheid.

The exploitation of nanostructuring to improve the performance of CMOS imagers based on microlens grids is a very promising avenue. In this perspective, numerical modeling is a key component to accurately characterize the absorption properties of these complex imaging structures, which are intrinsically multiscale (from the micrometer scale of the lenses to the nanometric characteristics of the nanostructured material layers). The FDTD (Finite Difference Time-Domain) method is the solution adopted in the first instance for the simulation of the interaction of light with this type of structure. However, because FDTD is based on a Cartesian mesh, this method shows limitations when it comes to accurately account for complex geometric features such as the curvature of microlenses or the texturing of material layer surfaces. In this context of the PhD of Jérémy Grebot, our main objectives are (1) to leverage locally refined meshes with a high order DGTD method for modeling the propagation of light in a nanostructured CMOS imager and, (2) to study and optimize the impact of nanostructuring for improving light absorption. In particular, for what concern the second objective, an inverse design methodology combining a high order DGTD method with a statistical learning-based global optimization algorithm is developed for studying various nanostructuring patterns of the surface of the absorbing semiconductor material such as one- and two-dimensional gratings.

6.2.1 Spectral analysis of nanostructures

Recent research in nanophotonics shows that the optical properties of nanostructures or metasurfaces can be derived in frequency-domain through the calculation of the supported optical resonances (see Fig. 1 (a)) also called Quasi-normal modes (QNM). Optical resonant modes are modeled by a nonlinear eigenvalue problem involving Maxwell’s equations, as a consequence the research devoted to solve such eigenvalue problem is a very active research field. In this work, we propose a comparative study on two recently introduced methods to solve nonlinear eigenvalue problems. In a first step, we discretize Maxwell’s equations with a finite element method (FEM) and we employ perfectly matched layers (PML) to account for the radiation condition at infinity. FEM generate very large matrices that turn eigenvalue equations solving in tedious tasks. Thus, in a second step, we implemented two advanced contour integral methods that can handle swiftly arbitrarily large matrices: the FEAST and Beyn algorithms. We then conducted a fine study of these two algorithms and the characteristics of different integration paths in the complex plane (see Fig. 1 (b)) to derive the eigenvalues and the associated eigenvectors of cylindrical particles with semi-analytic solutions. This study will ultimately allow us to determine the fastest and most accurate method for the computation of optical resonances on particles with arbitrary geometries. Such approach will provide a very good framework upon which optimization algorithms will be implemented to conduct research on photonic structures with optimal optical properties.

(a) Resonant modes of a cylinder made of GaAs in air. The radius of the cylinder is 500 nm the dielectric permittivity of GaAs and air are 14 and 1 respectively. The squared frequencies of the displayed eigenmodes are f1=8.95-0.4if_1=8.95-0.4i and f2=9.09-0if_2=9.09-0i, from left to right. (b) Map of the complex plane hosting the squared resonances of the structure described above. The blue crosses give the position of resonant modes and the green circle an exemple of contour to derive the modes displayed in panel (a).

6.2.2 Homogenization theory for metasurfaces

Participants: Agnès Maurel [ESPCI, Paris], Kim Pham [ENSTA, Paris], Nicolas Lebbe.

We applied the (surfacic) quasi-periodic homogeneization theory to derive equivalent transmission condition of deeply subwavelength microstructured metasurfaces. The resulting conditions are closely related to the so-called GSTC (Generalized Sheet Transition Conditions) that were originally derived in the 90s from physicial principles. Our approach provides a direct link between the coefficients of the susceptibility tensors driving surfacic polarization fields and the structural design of a microstructured metasurface. These equivalent boundary conditions are useful for numerical simulations since they reduce the complexity of the meshes at the interfaces and thus the number of degrees of freedom to be solved (see Fig. 2 (a)). The numerical implementation of these GSTCs was done both in 2D and 3D using the finite element method (FEM) in the frequency-domain (see Fig. 2 (b)). This year we have started to investigate the extension of this work in order to deal with resonant subwavelenth nanostructures, which is a prerequisite to design metasurfaces with enhanced light front control and shaping.

6.2.3 Equilibrated error estimators for high-order methods

Participants: Théohpile Chaumont-Frelet [SERENA project-team, Inria Paris], Alexandre Ern [SERENA project-team, Inria Paris], Patrick Vega [SERENA project-team, Inria Paris], Martin Vohralík [SERENA project-team, Inria Paris].

Error estimators are an important tool in the discretization of PDE problems. On the one hand, they permit to estimate the accuracy of the discrete model. On the other hand, estimators typically provide local information about the error in parts of the discretization mesh, enabling local refinement algorithms. A now well-established and widely used family of error estimators is called residual-based. These estimators have the advantage to be relatively easy to implement, and very efficient. Unfortunately, they become less accurate as the discretization order is increased. This leads in general to large error over-estimation, and decreased performance of adaptive algorithms.

In this collaboration with the SERENA project-team, we are interested in a modern family of error estimators, called equilibrated-fluxes. These estimators are slightly more costly to compute than residual-based ones, but they are much more efficient when combined with high-order discretization schemes. A key properties of such estimators is that they are p-robust, meaning that they do not suffer from an increase in the polynomial degree p employed in the discretization method. Our first key contribution in this topic that are concerned with the discretization of Helmholtz problems, has been published this year 12. Since 2020, we have focused on Maxwell's equations in the low-frequency limit. While the low-frequency limit is not directly of interest to the applications considered in the ATLANTIS project-team, it is mathematically interesting as a simplified foundation for later extension to the high-frequency case. Our first key results, where we introduce a broken patch-wise equilibrated estimators have been accepted for publication results 13. Further developments include the design of fully equilibrated estimators 32 as well as alternative equilibration strategies 27. Both of these works have been submitted for publication and are currently under review. Related theoretical results concerning approximation properties of Nédélec elements under minimal regularity have also been published 15.

6.2.4 Residual error estimators for high frequency problems

Participants: Théohpile Chaumont-Frelet, Stéphane Lanteri, Patrick Vega.

Residual error estimators are widely employed in applications to drive adaptive mesh refinement algorithms. However, most of the available current works in the literature deals with coercive problems and unfortunately, this setting does not cover high frequency wave problems, which are of central interest to the ATLANTIS project-team. This year, we started to investigate the use of residual error estimator for Maxwell's equations coupled with a linearized hydrodynamic Drude model. We covered the theoretical analysis and our two-dimensional numerical experiments are promising. This work has been accepted for publication 29. Some technical results concerning the approximability of high frequency solutions to Maxwell's equations have also been submitted 31.

6.2.5 Analysis of quasi-periodic boundary conditions

Participants: Théohpile Chaumont-Frelet, Zakaria Kassali, Stéphane Lanteri.

Quasi-perodidc boundary conditions arise in a number of applications considered in the team. Important examples are the simulations of solar cells and photonic crystals. These “unusual” boundary conditions correspond to an incident wave with an oblique incidence, and can strongly impact the stability and accuracy of numerical discretizations. We investigate these boundary conditions in the PhD of Zakaria Kassali. We have been able to derive sharp stability estimates for finite element discretizations involving quasi-periodicity. In particular, we have shown that a drastic accuracy loss can appear at certain angle of incidence, that mathematically correspond to “parabolically trapped” rays (see Fig. 3, blue curve). We propose to remedy this problem by using a multiscale hybrid-mixed (MHM) method (see Fig. 3, red curve). Our numerical investigations are rather promising, and current efforts are guided toward a rigorous error analysis for the MHM method, showing its robustness with respect to parabolically trapped rays.

6.2.6 Controllability methods

Participants: Théophile Chaumont-Frelet [University of Basel, Switzerland], Marcus Gröte [University of Basel, Switzerland], Stéphane Lanteri [University of Basel, Switzerland], Jet Tang [University of Basel, Switzerland].

The limiting amplitude principle predicts the asymptotic harmonic regime of a structure that is monochromatically illuminated. This makes a frequency-domain approach relevant, viewed as a long time asymptotic of a monochromatic time-domain simulation setting. Besides, it means that a time-domain solver may be used to solve frequency-domain problems, avoiding an explicit linear system solve. In this context, we investigate a modeling approach that allows to build bridges between the frequency- and time-domain approaches. This works heavily relies on the possibility of having reliable, accurate and high order discretization strategies for approximating time-domain problems. In collaboration with Marcus J. Grote and Jet Tang, University of Basel, Switzerland, we consider the use of time-domain solvers to efficiently solve frequency-domain problems in non dispersive materials. Specifically, we consider an approach called conjugate gradient controlability method that we analyze theoretically in 3D and numerically in 2D 28. We have shown that the proposed methods converges toward the time-harmonic solution, and numerical examples suggest that the method is robust in the high-frequency regime and/or in the presence of trapped waves in complex geometries.

6.2.7 Multiscale analysis of highly heterogeneous problems

Participants: Théophile Chaumont-Frelet [SERENA project-team, Inria Paris], Alexandre Ern [SERENA project-team, Inria Paris], Simon Lemaire [RAPSODI project-team, Inria Lille], Euan Spence [University of Bath, UK], Barbara Verfürth [Karlsruhe Institute for technology, Germany].

The analysis of multiscale highly heterogeneous problems where the characteristic size of the heterogeneities is much smaller than the wavelength is one of the core research topics of the ATLANTIS project-team. This year, we made three important contributions. (i) First, in collaboration with Euan Spence at University of Bath, UK, we derived frequency-explicit homogenization results for high-frequency Helmholtz problems 30. To the best of our knowledge, this work is the first contribution to rigorously established the convergence of the highly heterogeneous solution toward the homogenized one for high-frequency problems. (ii) Then, in collaboration with Alexandre Ern from the SERENA project-team and Simon Lemaire from the RAPSODI project-team, we established a bridge between two multiscale methods: the MHM and MsHHO methods 11. Specifically, we have shown that when the right-hand side is piecewise polynomial, the two methods produce the same solution, although they represent it with different degrees of freedom. We believe that this abstract result to be important, as it gives more insight about the two multiscale methods, and we hope it will provide several opportunities for improving them in the future. Finally (iii), with Barbara Verfürth at the Karlsruhe Institute for technology, Germany, we modified a mulitscale method known as LOD to handle sign-changing coefficients, which are crucial in the modeling of metamaterials 14.

6.3.1 Multiobjective statistical learning optimization of RGB metalens

Participants: Mickaël Binois [ACUMES project-team], Régis Duvigneau [ACUMES project-team], Patrice Genevet [CRHEA laboratory, Sophia Antipolis], Mahmoud Elsawy, Stéphane Lanteri.

Modeling of multi-wavelength metasurfaces relies on adjusting the phase of individual nanoresonators at several wavelengths. The most widely used design procedure neglects the near-field coupling between the nanoresonators, which dramatically reduces the overall diffraction efficiency, bandwidth, numerical aperture, and device diameter. Another alternative design strategy is to combine a numerical optimization technique with fullwave simulations to mitigate this problem and optimize the entire metasurface at once. In this work, we have developed a global multiobjective optimization technique that leverages a statistical learning method to optimize RGB spherical metalenses at visible wavelengths. The optimization procedure, coupled to a fullwave solver based on a high order DGTD method, accounts for the near-field coupling between the resonators as indicated in Fig. 4 (a). High numerical aperture RGB metalenses (numerical aperture $=0.47$ and $0.56$) of 8 and 10 $\mu$m diameters are optimized with numerical average focusing efficiencies of $55\%$ and $45\%$, respectively, and with an average focusing error of less than $6\%$ for the RGB colors as it is shown in Fig. 4 (c). The fabricated and experimentally characterized devices (see Fig. 4 (b)) yield $44.16\%$ and $31.5\%$ respective efficiencies. The reported performances represent the highest focusing efficiencies for high numerical aperture $>0.5$ RGB metalenses obtained so far. The integration of multi-wavelength metasurfaces in portable and wearable electronic devices requires high performances to offer a variety of applications ranging from classical imaging to virtual and augmented reality. The manuscript has been published in one of the most prestigious scientific journals (ACS photonics) in the field of nanophotonics and metasurface 17.

       

6.3.2 Optimization of metasurfaces under geometrical uncertainty using statistical learning

Participants: Mickaël Binois [ACUMES project-team, Inria Sophia Antipolis-Méditerranée],, Régis Duvigneau [ACUMES project-team, Inria Sophia Antipolis-Méditerranée],, Patrice Genevet [CRHEA laboratory, Sophia Antipolis], Mahmoud Elsawy, Stéphane Lanteri.

The fabrication process of metasurface components requires dealing with very small sizes in the nanometer scale. Despite the revolutionary nanofabrication facilities developed in the past few years, uncertainties and systematic fabrication errors are usually non-negligible and are the source of efficiency degradation compared to modeling and simulation results. The performance of metasurfaces measured experimentally often discords with expected values from numerical optimization. These discrepancies are attributed to the poor tolerance of metasurface building blocks concerning fabrication uncertainties and nanoscale imperfections. Quantifying their efficiency drop according to geometry variation is crucial to improve the range of application of this technology. In this work, we have introduced a novel optimization methodology to account for the manufacturing errors related to metasurface designs. In this approach, accurate results using probabilistic surrogate models are used to reduce the number of costly numerical simulations. Our approach necessitates resolving a bi-objective optimization problem that accounts for the mean and the variance change of the efficiency under the given noise. With the UQ analysis, we obtained designs that are twice more robust than the analogous one in the deterministic case. We employ our procedure to optimize the classical beam steering metasurface made of cylindrical nanopillars as shown in Fig. 5. Our numerical results reveal that the performance of the optimized design taking into account the UQ analysis (Fig. 5 (b)) is much more robust compared to the optimized design in the deterministic case (Fig. 5 (a)). In other words, perturbing the diameters of the pillars in the deterministic case by $\pm 12$ nm dramatically reduces the overall efficiency (our target wavelength is 600 nm). Yet, the efficiency of the optimized design in the UQ case remains almost constant under the same diameter perturbations. This work has been published in Optics Express, which is a well-known open-access journal in the field of nanophotonics 16.

Comparison between the performance of the best design obtained in the deterministic case (a) (without taking into account the UQ) and the best design obtained from our UQ optimization methodology (b). In each case we plot the design without any noise (blue curves in each case) and trace the performance when adding ±12\pm 12 nm to the optimized diameters. The optimized geometry in each case is shown in the insets.

6.3.3 Nanoimprint color filter metasurface for imaging devices

Participants: David Grosso [Institut Matériaux Microélectronique Nanosciences de Provence (IM2NP) Aix-Marseille Université and Solnil], Marco Abbarchi [Institut Matériaux Microélectronique Nanosciences de Provence (IM2NP) Aix-Marseille Université and Solnil], Badre Kerzabi [Solnil], Mahmoud Elsawy, Alexis Gobé, Stéphane Lanteri.

Due to the rapid growth of metasurface applications, several manufacturing techniques have been developed in the past few years. Among them, NanoImprint Lithography (NIL) has been considered as a promising fabrication possibility for metasurface. NIL can achieve features below 100 nm without the need for complex and expensive optics and light sources needed to achieve similar resolutions in advanced photolithography. Yet, despite the advantages of this fabrication procedure, it has some limitations related to the dimensions of the single nanoresonators. In other words, the fabrication constraints related to the aspect ratio (ratio between height and width) is important. Typically, a maximum aspect ratio of 2 can be considered in such a technique. In this work, together with our collaborators from IM2NP, Aix-Marseille Université and the Solnil startup who are specialists in nanoimprint technology, we aim at optimizing a low cost colour filtering metasurface for imaging applications taking into account all the fabrication constraints. The main goal of this study is to deploy our optimization methodology to maximize the efficiency of such metasurface configuration. Various shapes with different physical mechanisms have been investigated. Fig. 6 illustrates one of the optimized designs based on cylindrical nanopillars (see caption for more details). It is worth mentioning that in this study various shapes have been optimized and studied numerically, yet, as a starting point for the fabrication, the cylindrical pillars will be considered as a first illustration for the nanoimprint fabrication.

6.3.4 Simulation of 3D conformal metasurfaces using GSTCs

Participants: Sandeep Yadav Golla [CRHEA laboratory, Sophia Antipolis], Patrice Genevet [CRHEA laboratory, Sophia Antipolis], Nicolas Lebbe.

The development of metamaterials and the associated technologies have exploited transformation optics and its analogy with light propagation in curved space to manipulate light in arbitrary ways. In this collaboration, we addressed the problem of metasurface tranmission conditions at conformal interfaces using Generalized Sheet Transitions Conditions (C-GSTCs) derived following pioneering works of Idemen for planar interfaces using surfacic Dirac distributions. We also adapted the classical inversion procedure used in planar GSTCs to retrieve the susceptibilities to the conformal case by restricting the tensors values to their sole tangential components. These C-GSTCs were first implemented in both 2D and 3D using the FEM (see Fig. 17) and then used to study the impact of the interface geometry on different types of aberrations for metalenses.

6.3.5 Optimization of light trapping in nanostructured solar cells

Participants: Stéphane Collin [Sunlit team, C2N-CNRS, Paris-Saclay], Alexis Gobé, Stéphane Lanteri.

There is significant recent interest in designing ultrathin crystalline silicon solar cells with active layer thickness of a few micrometers. Efficient light absorption in such thin films requires both broadband antireflection coatings and effective light trapping techniques, which often have different design considerations. In collaboration with physicists from the Sunlit team at C2N-CNRS, We exploit statistical learning methods for the inverse design of material nanostructuring with the goal of optimizing light trapping properties of ultraphin solar cells. This objective is challenging because the underlying electromagnetic wave problems exhibit multiple resonances, while the geometrical settings are non-trivial. Such multi-resonant solar cell structures are attractive for maximizing light absorption for the full solar light spectrum as illustrated in Fig. 8. We exploit statistical learning methods for the inverse design of material nanostructuring with the goal of optimizing light trapping properties of ultraphin solar cells. This study is conducted in collaboration with the Sunlit headed by Stéphane Collin at the Center for Nanoscience and Nanotechnology (C2N, CNRS).

6.3.6 Plasmonic sensing with nanocubes

Participants: Antoine Moreau [Institut Pascal, Clermont-Ferrand], Stéphane Lanteri, Guillaume Leroy, Claire Scheid.

The propagation of light in a slit between metals is known to give rise to guided modes. When the slit is of nanometric size, plasmonic effects must be taken into account, since most of the mode propagates inside the metal. Indeed, light experiences an important slowing-down in the slit, the resulting mode being called gap-plasmon. Hence, a metallic structure presenting a nanometric slit can act as a light trap, i.e. light will accumulate in a reduced space and lead to very intense, localized fields. We study the generation of gap plasmons by various configurations of silver nanocubes separated from a gold substrate by a dielectric layer, thus forming a narrow slit under the cube. When excited from above, this configuration is able to support gap-plasmon modes which, once trapped, will keep bouncing back and forth inside the cavity. We exploit statistical learning methods for the goal-oriented inverse design of cube size, dielectric and gold layer thickness, as well as gap size between cubes in a dimer configuration. This study is conducted in collaboration with Antoine Moreau at Institut Pascal (CNRS).

6.3.7 Multiple scattering in random media

Participants: Stéphane Descombes [INPHYNI laboratory, Sophia Antipolis], Stéphane Lanteri [INPHYNI laboratory, Sophia Antipolis], Guillaume Leroy [INPHYNI laboratory, Sophia Antipolis], Cédric Legrand [INPHYNI laboratory, Sophia Antipolis], Gian Luca Lippi [INPHYNI laboratory, Sophia Antipolis].

Fluorescence signals emitted by probes, used to characterize the expression of biological markers in tissues or cells, can be very hard to detect due to a small amount of molecules of interest (proteins, nucleic sequences), to specific genes expressed at the cellular level, or to the limited number of cells expressing these markers in an organ or a tissue. Access to information coming from weaker emitters can only come from strengthening the signal, since electronic post-amplification raises the noise floor as well. While costly and molecule-specific biochemical processes are being developed for this purpose, a new mechanism based on the simultaneous action of stimulated emission and multiple scattering, induced by nanoparticles suspended in the sample, has been recently demonstrated to effectively amplify weak fluorescence signals. A precise assessment of the signal fluorescence amplification that can be achieved by such a scattering medium requires a electromagnetic wave propagation modeling approach capable of accurately and efficiently coping with multiple space and time scales, as well as with non-trivial geometrical features (shape and topological organization of scatterers in the medium). In the context of a collaboration with physicists from the Institut de Physique de Nice INPHYNI (Gian Luca Lippi from the complex photonic systems and materials group), we initiated this year a study on the simultaneous action of stimulated emission and multiple scattering by randomly distributed nanospheres in a bulk medium. From the numerical modeling point of view, or short term goal is to develop a time-domain numerical methodology for the simulation of random lasing in a gain medium.

8.1.1 STIC/MATH/CLIMAT AmSud project

8.2.1 Visits of international scientists

8.3.1 FP7 & H2020 projects

8.4.1 ANR project

9.1.1 Scientific events: organisation

9.1.2 Journal

9.2.1 Teaching

• Claire Scheid, Analyse Fonctionnelle, Practical works, Master 1 MPA, 18 h, Université Côte d'Azur
• Claire Scheid, Optimisation et éléments finis, Master 1 MPA and IM, 54 h, Université Côte d'Azur
• Claire Scheid, Option Modélisation, Lectures and practical works, Master 2 Agrégation, 43 h, Université Côte d'Azur
• Yves D'Angelo, Modélisation Simulation Numérique, Master 1 IM/MPA, 50 h, Université Côte d'Azur
• Yves D'Angelo, Modélisation de la Turbulence dans l'Industrie, Master 1 MPA and IM, 16 h, Université Côte d'Azur
• Stéphane Descombes, Introduction to partial differential equations Master 1 MPA and IM, 30 h, Université Côte d'Azur
• Stéphane Descombes, Principal component analysis and supervised analysis, Master 2 SIRIS, Université Côte d'Azur
• Stéphane Descombes, Mathematics practical works, L1 SocioEco, 18 h, Université Côte d'Azur
• Engineering: Stéphane Lanteri, High performance scientific computing, 24 h, MAM5, Polytech Nice Sophia - Université Côte d'Azur

9.2.2 Supervision

• PhD in progress: Jérémy Grebot, Advanced modeling of nanostructured CMOS imagers, Stéphane Lanteri and Claire Scheid
• PhD in progress: Zakaria Kassali, Multiscale finite element simulations applied to the design of photovoltaic cells, Théophile Chaumont-Frelet and Stéphane Lanteri
• PhD in progress: Cédric Legrand, High order finite element type solvers for modeling gain media, Stéphane Descombes and Stéphane Lanteri
• PhD in progress: Thibault Laufroy, High order finite element type solvers for thermoplamonics, Yves d'Angelo and Claire Scheid
• PhD in progress: Massimiliano Montone, High order finite element type solvers for the coupled Maxwell-semiconductor equations in the time-domain, Stéphane Lanteri and Claire Scheid

9.2.3 Juries

• Claire Scheid, examiner, Joe Alhelou, Institut de Mathématiques de Toulouse, Université de Toulouse, France, November 2021
• Stéphane Lanteri, examiner, Claire Scheid, HDR defense, Université Côte d'Azur, December 2021
• Stéphane Lanteri, reviewer, Kassem Asfour, Université de Toulouse, France, November 2021
• Stéphane Lanteri, examiner, Sandeep Yadav, Golla, CRHEA, Université Côte d'Azur, November 2021
• Stéphane Lanteri, examiner, Anthony Gourdin, Université de Montpellier, May 2021

9.2.4 Internal or external Inria responsibilities

• Yves D'Angelo: head of the J.A. Dieudonné Mathematics Laboratory
• Stéphane Descombes: director of the “Maison de la Modélisation, de la Simulation et des Interactions” (MSI) at Université Côte d'Azur
• Stéphane Lanteri: member of the Project-team Committee's Bureau of the Inria research center at Université Côte d'Azur
• Stéphane Lanteri: member of the Scientific Committee of Cerfacs
• Stéphane Lanteri: member of the Scientific Committee of the Wavecomplexity initiative at Université Côte d'Azur
• Stéphane Lanteri: member of the “Comité de Chercheurs Calculant au CINES”

International journals

• 10 articleC.Camille Carvalho, P.Patrick Ciarlet and C.Claire Scheid. Limiting amplitude principle and resonances in plasmonic structures with corners: numerical investigation.Computer Methods in Applied Mechanics and EngineeringOctober 2021
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• 11 articleT.Théophile Chaumont-Frelet, A.Alexandre Ern, S.Simon Lemaire and F.Frédéric Valentin. Bridging the Multiscale Hybrid-Mixed and Multiscale Hybrid High-Order methods.ESAIM: Mathematical Modelling and Numerical Analysis2021
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• 12 articleT.Théophile Chaumont-Frelet, A.Alexandre Ern and M.Martin Vohralík. On the derivation of guaranteed and p-robust a posteriori error estimates for the Helmholtz equation.Numerische Mathematik1483April 2021, 525-573
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• 13 articleT.Théophile Chaumont-Frelet, A.Alexandre Ern and M.Martin Vohralík. Stable broken H(curl) polynomial extensions and p-robust a posteriori error estimates by broken patchwise equilibration for the curl-curl problem.Mathematics of ComputationSeptember 2021
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• 14 articleT.Théophile Chaumont-Frelet and B.Barbara Verfürth. A generalized finite element method for problems with sign-changing coefficients.ESAIM: Mathematical Modelling and Numerical Analysis553May 2021, 939-967
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• 15 articleT.Théophile Chaumont-Frelet and M.Martin Vohralík. Equivalence of local-best and global-best approximations in H(curl).Calcolo58November 2021, 53
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• 16 articleM.Mahmoud Elsawy, M.Mickaël Binois, R.Régis Duvigneau, S.Stéphane Lanteri and P.Patrice Genevet. Optimization of metasurfaces under geometrical uncertainty using statistical learning.Optics Express29192021, 29887
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• 17 articleM. M.Mahmoud M R Elsawy, A.Anthony Gourdin, M.Mickaël Binois, R.Régis Duvigneau, D.Didier FELBACQ, S.Samira Khadir, P.Patrice Genevet and S.Stéphane Lanteri. Multiobjective statistical learning optimization of RGB metalens.ACS photonics88July 2021, 2498–2508
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• 18 articleK.Kun Li, T.-Z.Ting-Zhu Huang, L.Liang Li and S.Stéphane Lanteri. Non-intrusive reduced-order modeling of parameterized electromagnetic scattering problems using cubic spline interpolation.Journal of Scientific Computing872May 2021
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Conferences without proceedings

• 19 inproceedingsT.Théophile Chaumont-Frelet, A.Alexandre Ern, S.Simon Lemaire and F.Frédéric Valentin. Bridging the multiscale hybrid-mixed and multiscale hybrid high-order methods.GS21 - SIAM Conference on Mathematical and Computational Issues in the GeosciencesMilan / Virtual, ItalyJune 2021
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• 20 inproceedingsT.Théophile Chaumont-Frelet, A.Alexandre Ern, S.Simon Lemaire and F.Frédéric Valentin. Bridging the multiscale hybrid-mixed and multiscale hybrid high-order methods.ICOSAHOM 2020 - International Conference on Spectral and High-Order MethodsVienna / Virtual, AustriaJuly 2021
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• 21 inproceedingsT.Théophile Chaumont-Frelet, A.Alexandre Ern and M.Martin Vohralík. Guaranteed error estimates for finite element discretizations of Helmholtz problems.Numerical WavesNice, FranceOctober 2021
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• 22 inproceedingsT.Théophile Chaumont-Frelet, A.Alexandre Ern and M.Martin Vohralík. Polynomial-degree-robust a posteriori error estimation for Maxwell's equations.WCCM & ECCOMAS Congress 2020 - 14th World Congress in Computational Mechanics and ECCOMAS CongressParis, FranceJanuary 2021
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• 23 inproceedingsT.Théophile Chaumont-Frelet, A.Alexandre Ern and M.Martin Vohralík. Polynomial-degree-robust a posteriori error estimation for Maxwell's equations.ICOSAHOM 2021 - International Conference on Spectral High Order MethodsVienna, AustriaJuly 2021
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• 24 inproceedingsT.Théophile Chaumont-Frelet and M.Martin Vohralík. An equilibrated a posteriori error estimator for the curl-curl problem.EFEF 2021 - 18th European Finite Element FairParis, FranceSeptember 2021
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• 25 inproceedingsM. M.Mahmoud M R Elsawy, M.Mickaël Binois, R.Régis Duvigneau, S.Stéphane Lanteri and P.Patrice Genevet. Statistical learning multiobjective optimization for large-scale achromatic metalens at visible regime.CLEO, Laser Science to Photonic ApplicationsSan Jose, California (web conference format), United StatesMay 2021
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• 26 inproceedingsM. .Mahmoud M. R. Elsawy, S.Stephane Lanteri, R.Régis Duvigneau and P.Patrice Genevet. Statistical Learning Optimization for Highly Efficient Metasurface Designs.SIAM Conference on Computational Science and Engineering 2021Texas, United StatesMarch 2021
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Reports & preprints

• 27 miscT.Théophile Chaumont-Frelet. A simple equilibration procedure leading to polynomial-degree-robust a posteriori error estimators for the curl-curl problem.August 2021
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• 28 miscT.Théophile Chaumont-Frelet, M. J.M J Grote, S.S Lanteri and J. H.J H Tang. A controllability method for Maxwell's equations.June 2021
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• 29 miscT.Théophile Chaumont-Frelet, S.Stéphane Lanteri and P.Patrick Vega. A posteriori error estimates for finite element discretizations of time-harmonic Maxwell's equations coupled with a non-local hydrodynamic Drude model.March 2021
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• 30 miscT.Théophile Chaumont-Frelet and E. A.Euan A Spence. Scattering by finely-layered obstacles: frequency-explicit bounds and homogenization.September 2021
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• 31 miscT.Théophile Chaumont-Frelet and P.Patrick Vega. Frequency-explicit approximability estimates for time-harmonic Maxwell's equations.May 2021
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• 32 miscT.Théophile Chaumont-Frelet and M.Martin Vohralík. p-robust equilibrated flux reconstruction in H(curl) based on local minimizations. Application to a posteriori analysis of the curl-curl problem.May 2021
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• 33 miscN.Nicolas Lebbe, K.Kim Pham and A.Agnès Maurel. Stable GSTC formulation for Maxwell’s equations.April 2021
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