Publications

2026

• Asymptotic models for time-domain scattering by small particles
  
  • Savchuk Adrian
  , 2026. In this manuscript, we address the problem of time-dependent wave scattering by multiple small particles of arbitrary shape. To approximate the solution of the associated boundary-value problem, we derive an asymptotic model that achieves a higher convergence rate compared to existing models. Our method relies on a boundary integral formulation, semi-discretized in space using a Galerkin approach, where the main challenge lies in choosing appropriate basis functions for the Galerkin space. We show that using equilibrium densities as basis functions yields a cubic-convergent asymptotic model that ensures a priori stability in the time domain due to the coercivity properties of the single-layer boundary integral operator. Unlike the case of spheres, where entries can be computed explicitly, the generalized method requires double integration, which becomes computationally expensive as the number of particles increases. To address this, we derive a simplified model that preserves the stability and convergence properties of the original formulation, alongside a lower-order but computationally advantageous Born model, providing a rigorous theoretical error analysis for both. When the distance between obstacles decreases with the small parameter, the cubic convergence of the scattered field can no longer be guaranteed. To overcome this limitation, we derive high-order asymptotic models by enriching the basis functions in the Galerkin space to maintain accuracy even for closely clustered particles. Finally, the proposed framework is applied to more complex interactions between a large obstacle and multiple small particles, and is further extended to time-domain electromagnetic scattering by small spheres. Numerical experiments validate the theoretical stability and performance across all proposed models.
• Metamaterials and Fluid Flows
  
  • Avallone Francesco
  • Bosia Federico
  • Chen Yi
  • Colombo Giada
  • Craster Richard
  • de Ponti Jacopo Maria
  • Fabbiane Nicolò
  • Haberman Michael
  • Hussein Mahmoud
  • Hwang Wontae
  • Iemma Umberto
  • Juhl Abigail
  • Kadic Muamer
  • Kotsonis Marios
  • Laude Vincent
  • Marquet Olivier
  • Mery Fabien
  • Michelis Theodoros
  • Nouh Mostafa
  • Ragni Daniele
  • Touboul Marie
  • Wegener Martin
  • Krushynska Anastasiia
  Nature Communications, Nature Publishing Group, 2026. (10.1038/s41467-026-70163-2)
  DOI : 10.1038/s41467-026-70163-2
• Discretization in multilayered media with high contrasts: is it all about the boundaries?
  
  • Carvalho Camille
  • Chaillat Stéphanie
  • Tsogka Chrysoula
  • Cortes Elsie A
  , 2026. Wave propagation in multilayered media with high material contrasts poses significant numerical challenges, as large variations in wavenumbers lead to strong reflections and complex transmission of the incoming wave field. To address these difficulties, we employ a boundary integral formulation thereby avoiding volumetric discretization. In this framework, the accuracy of the numerical solution depends strongly on how the material interfaces are discretized. In this work, we demonstrate that standard meshing strategies based on resolving the maximum wavenumber across the domain become computationally inefficient in multilayered configurations, where high wavenumbers are confined to localized subdomains. Through a systematic study of multilayer transmission problems, we show that no simple discretization rule based on the maximum wavenumber or material contrasts emerges. Instead, the wavenumber of the background (exterior) medium plays a dominant role in determining the optimal boundary resolution. Building on these insights, we propose an adaptive approach that achieves uniform accuracy and efficient computation across multiple layers. Numerical experiments for a range of multilayer configurations demonstrate the scalability and robustness of the proposed approach.
• Htool-DDM: A C++ library for parallel solvers and compressed linear systems.
  
  • Marchand Pierre
  • Tournier Pierre-Henri
  • Jolivet Pierre
  Journal of Open Source Software, Open Journals, 2026, 11 (118), pp.9279. (10.21105/joss.09279)
  DOI : 10.21105/joss.09279
• Automated far-field sound field estimation combining robotized acoustic measurements and the boundary elements method
  
  • Pascal Caroline
  • Marchand Pierre
  • Chapoutot Alexandre
  • Doaré Olivier
  Acta Acustica, EDP Sciences, 2026. The identification and reconstruction of acoustic fields radiated by unknown structures is usually performed using either Sound Field Estimation or Near-field Acoustic Holography techniques. The latter turns out to be especially useful when data is only available close to the source, but information throughout the whole space is needed. Yet, the lack of amendable and efficient implementations of state-of-the-art solutions, as well as the laborious and often lengthy deployment of acoustic measurements continue to be significant obstacles to the practical application of such methods. The purpose of this work is to address both problems. First, a completely automated metrology setup is proposed, in which a robotic arm is used to gather extensive and accurately positioned acoustic data without any human intervention. The impact of the robot on acoustic pressure measurements is cautiously evaluated, and proved to remain limited below 1 kHz. The Sound Field Estimation is then tackled using the Boundary Element Method, and implemented using the FreeFEM software. Numerically simulated measurements have allowed us to assess the method accuracy, which matches theoretically expected results and proves to remain robust against positioning inaccuracies, provided that the robot is carefully calibrated. The overall solution has been successfully tested using actual robotized measurements of an unknown loudspeaker, with a reconstruction error of less than 30 %. (10.1051/aacus/2026017)
  DOI : 10.1051/aacus/2026017
• Asymptotic analysis at any order of Helmholtz's problem in a corner with a thin layer: an algebraic approach
  
  • Baudet Cédric
  Asymptotic Analysis, IOS Press, 2026. We consider the Helmholtz equation in an angular sector partially covered by a homogeneous layer of small thickness, denoted ε. We propose in this work an asymptotic expansion of the solution with respect to ε at any order. This is done using matched asymptotic expansion, which consists here in introducing different asymptotic expansions of the solution in three subdomains: the vicinity of the corner, the layer and the rest of the domain. These expansions are linked through matching conditions. The presence of the corner makes these matching conditions delicate to derive because the fields have singular behaviors. Our approach is to reformulate these matching conditions purely algebraically by writing all asymptotic expansions as formal series. By using algebraic calculus we reduce the matching conditions to scalar relations linking the singular behaviors of the fields. These relations have a convolutive structure and involve some coefficients that can be computed analytically. Our asymptotic expansion is justified rigorously with error estimates. (10.1177/09217134251389983)
  DOI : 10.1177/09217134251389983
• Wave propagation in the frequency regime in one-dimensional quasiperiodic media -Limiting absorption principle
  
  • Amenoagbadji Pierre
  • Fliss Sonia
  • Joly Patrick
  , 2026. <div><p>We study the one-dimensional Helmholtz equation with (possibly perturbed) quasiperiodic coefficients. Quasiperiodic functions are the restriction of higher dimensional periodic functions along a certain (irrational) direction. In classical settings, for real-valued frequencies, this equation is generally not well-posed: existence of solutions in L 2 is not guaranteed and uniqueness in L ∞ may fail. This is a well-known difficulty of Helmholtz equations, but it has never been addressed in the quasiperiodic case. We tackle this issue by using the limiting absorption principle, which consists in adding some imaginary part (also called absorption) to the frequency in order to make the equation well-posed in L 2 , and then defining the physically relevant solution by making the absorption tend to zero. In previous work, we introduced a definition of the solution of the equation with absorption based on Dirichlet-to-Neumann (DtN) boundary conditions. This approach offers two key advantages: it facilitates the limiting process and has a direct numerical counterpart. In this work, we first explain why the DtN boundary conditions have to be replaced by Robin-to-Robin boundary conditions to make the absorption go to zero. We then prove, under technical assumptions on the frequency, that the limiting absorption principle holds and we propose a numerical method to compute the physical solution.</p></div>
• Discrete FEM-BEM coupling with the Generalized Optimized Schwarz Method
  
  • Boisneault Antonin
  • Bonazzoli Marcella
  • Claeys Xavier
  • Marchand Pierre
  , 2026. The present contribution aims at developing a non-overlapping Domain Decomposition (DD) approach to the solution of acoustic wave propagation boundary value problems based on the Helmholtz equation, on both bounded and unbounded domains. This DD solver, called Generalized Optimized Schwarz Method (GOSM), is a substructuring method, that is, the unknowns of an iteration are associated with the subdomains interfaces. We extend the analysis presented in a previous paper of one of the author to a fully discrete setting. We do not consider only a specific set of boundary conditions, but a whole class including, e.g., Dirichlet, Neumann, and Robin conditions. Our analysis will also cover interface conditions corresponding to a Finite Element Method - Boundary Element Method (FEM-BEM) coupling. In particular, we shall focus on three classical FEM-BEM couplings, namely the Costabel, Johnson-Nédélec and Bielak-MacCamy couplings. As a remarkable outcome, the present contribution yields well-posed substructured formulations of these classical FEM-BEM couplings for wavenumbers different from classical spurious resonances. We also establish an explicit relation between the dimensions of the kernels of the initial variational formulation, the local problems and the substructured formulation. That relation especially holds for any wavenumber for the substructured formulation of Costabel FEM-BEM coupling, which allows us to prove that the latter formulation is well-posed even at spurious resonances. Besides, we introduce a systematically geometrically convergent iterative method for the Costabel FEM-BEM coupling, with estimates on the convergence speed.
• Early-Reverberation Imaging Functions for Bounded Elastic Domains
  
  • Ducasse Eric
  • Rodriguez Samuel
  • Bonnet Marc
  Acta Acustica, EDP Sciences, 2026, 10, pp.2. For the ultrasonic inspection of bounded elastic structures, finite-duration imaging functions are derived in the Fourier-Laplace domain.The signals involved are exponentially windowed, so that early reflections are taken into account more strongly than later ones in the imaging methodology.Applying classical approaches to the general case of anisotropic elasticity, we express the Fréchet derivatives of the relevant data-misfit functional with respect to arbitrary perturbations of the mass density and stiffnesses in terms of forward and adjoint solutions.Their definitions incorporate the exponentially decaying weighting. The proposed finite-duration imaging functions are then defined on that basis.As some areas of the structure are less insonified than others, it is necessary to define normalized imaging functions to compensate for these variations.Our approach in particular aims to overcome the difficulty of dealing with bounded domains containing defects not located in direct line of sight from the transducers and measured signals of long duration.For this initiation work, we demonstate the potential of the proposed method on a two-dimensional test case featuring the imaging of mass and elastic stiffness variations in a region of a bounded isotropic medium that is not directly visible from the transducers. (10.1051/aacus/2025069)
  DOI : 10.1051/aacus/2025069
• Aeroacoustic Optimization of Transonic Open Rotors Using a Discrete Adjoint Framework
  
  • Mohammedi Yacine
  • Daroukh Majd
  • Buszyk Martin
  • Hajczak Antoine
  • Bonnet Marc
  • Salah El-Din Itham
  AIAA Journal, American Institute of Aeronautics and Astronautics, 2026. A discrete adjoint framework is developed to optimize rotor aerodynamic performance and tonal noise from steady-state simulations in the rotating frame of reference. Aerodynamic performance is characterized through the thrust and the torque, while the acoustic metric is determined using a simplified expression of the off-blade frequency-domain Ffowcs-Williams and Hawkings (FW-H) equation for far-field observers, following the model of Hanson and Parzych (1993) originally formulated for on-blade surfaces. This formulation is implemented and validated against an established time-domain FW-H solver.~Analytical expressions are derived for the sensitivities of aerodynamic quantities and far-field acoustic pressure, which enables the reconstruction of any objective function expressed in terms of acoustic pressure, such as acoustic power. These analytical sensitivities are successfully validated by comparison to second-order accurate finite-difference (FD) approximations. A mesh deformation tool updates the geometry and computes the corresponding gradient of the volume mesh. The discrete adjoint of a Reynolds-Averaged Navier–Stokes solver is then used to provide the sensitivities of the objective functions with respect to the blade shape parameters. These sensitivities are supplied to the optimizer, which updates the design parameters accordingly, and the mesh deformation tool subsequently modifies the geometry of the blade based on the optimizer’s feedback, thereby completing the gradient-based aeroacoustic design workflow. The optimization targets a composite objective function that combines torque and acoustic power at the Blade Passing Frequency (BPF), while enforcing thrust as a constraint. The whole process is applied on an academic isolated open rotor configuration at transonic conditions with equal weighting assigned to aerodynamic and acoustic objectives, resulting in a combined reduction of approximately 7% in torque and 13dB in acoustic power. Unsteady assessments of both the isolated rotor and rotor-stator configurations further show that the optimization reduces the dominant rotor noise without increasing the second and third BPF levels, while the remaining unsteady rotor and rotor-stator interaction sources represent opportunities for further noise reduction.
• Boundary integral equation and convolution quadrature methods for transient problems: an introduction
  
  • Bonnet Marc
  • Chaillat Stéphanie
  , 2026. Boundary integral equation (BIE) methods are well-suited for the numerical solution of a wide range of linear PDEs, including evolution problems arising in acoustics, elastodynamics, electromagnetism, heat transfer, and related fields. While time-domain BIE formulations have a long history, the development of efficient numerical methods for their solution is more recent. Classical time-stepping BIE methods based on retarded potentials raise well-documented issues, which prompted the more-recent emergence of alternative approaches based on the concept of convolution quadrature (CQ). This review paper aims at providing an accessible and tutorialstyle presentation of CQ-based methods for time-domain BIEs. These exist in two main forms, namely the sequential-in-time CQ-BEM and a recent simultaneous-in-time form herein referred to as the Z-BEM. Both forms are based on adopting at the outset a time-discrete setting. Consequently, as informally shown in this paper, both approaches may be derived and described in an unified manner by means of the Z-transform, a discrete-time analog of the Laplace transform, with the linear evolution problem to be solved initially set either in PDE form or as a continuous time-domain BIE. Computational aspects, including algorithmic parameters and high-frequency approximations, are also discussed, together with comparative remarks on the Z-BEM and CQ-BEM approaches and a concise survey of the main developments and applications.
• Multiscale modeling for a class of high-contrast heterogeneous sign-changing problems
  
  • Chung Eric T.
  • Ciarlet Patrick
  • Jin Xingguang
  • Ye Changqing
  Journal of Computational Mathematics -International Edition-, Global Science Press, 2026. The mathematical formulation of sign-changing problems involves a linear second-order partial differential equation in the divergence form, where the coefficient can assume positive and negative values in different subdomains. These problems find their physical background in negative-index metamaterials, either as inclusions embedded into common materials as the matrix or vice versa. In this paper, we propose a numerical method based on the constraint energy minimizing generalized multiscale finite element method (CEMGMsFEM) specifically designed for sign-changing problems. The construction of auxiliary spaces in the original CEM-GMsFEM is tailored to accommodate the sign-changing setting. The numerical results demonstrate the effectiveness of the proposed method in handling sophisticated coefficient profiles and the robustness of coefficient contrast ratios. Under several technical assumptions and by applying the T-coercivity theory, we establish the inf-sup stability and provide an a priori error estimate for the proposed method.
• Hybrid FEM/IPDG semi-implicit schemes for time domain electromagnetic wave propagation in non cylindrical coaxial cables
  
  • Beni Hamad Akram
  • Imperiale Sébastien
  • Joly Patrick
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2026. In this work, we develop an efficient numerical method for solving 3D Maxwell's equations in non-cylindrical coaxial cables. The main challenge arises from the elongated geometry of the computational domain, which induces strong anisotropy between the longitudinal direction (along the cable) and the transverse directions (within the cross-sections). This leads to the use of highly anisotropic meshes, where the longitudinal mesh size is much larger than the transverse one.<p>Our objective is to design a numerical scheme that is explicit in the longitudinal direction, with a CFL stability condition depending only on the longitudinal mesh size. In a previous work, we achieved this for cylindrical cables by employing prismatic edge elements, 1D quadrature for longitudinal mass lumping, and a hybrid explicit/implicit time discretization. The present paper extends this approach to non-cylindrical cables, addressing several new difficulties with the following key ingredients: (1) representing the cable as a deformation of a reference cylindrical cable and employing mapping techniques between the physical and reference domains; (2) using an anisotropic space discretization that combines an interior penalty discontinuous Galerkin (IPDG) method in the transverse directions with a conforming finite element method in the longitudinal direction; (3) utilizing prismatic edge elements on a prismatic mesh of the reference cable; and (4) adapting the construction of the hybrid explicit-implicit time discretization to the new structure of the semidiscrete problem. From a theoretical perspective, the main difficulty lies in the stability analysis, which requires extending and adapting standard techniques for DG methods in space and energy methods in time.</p>
• Convergence analysis of GMRES applied to Helmholtz problems near resonances
  
  • Dolean Victorita
  • Marchand Pierre
  • Modave Axel
  • Raynaud Timothée
  Journal of Computational and Applied Mathematics, Elsevier, 2026. The finite element solution of Helmholtz problems near resonant or quasi-resonant frequencies poses significant challenges, as iterative solvers typically suffer from severely degraded convergence. We analyze the convergence behavior of GMRES applied to linear systems arising from such configurations. Theoretical convergence estimates are derived based on harmonic Ritz values, highlighting their proximity to small eigenvalues as a key determining factor. We further examine deflation strategies and their interplay with preconditioning techniques, using the Complex Shifted Laplacian preconditioner as a case study. Numerical experiments on resonant and quasi-resonant test cases validate the theoretical framework and demonstrate the effectiveness of deflation strategies. This study provides new insights and practical guidance for analyzing and improving iterative solvers for time-harmonic problems near resonances.
• A general-purpose global regularization method for 3D volume integral operators
  
  • Anderson Thomas G.
  • Bonnet Marc
  • Faria Luiz M.
  • Pérez-Arancibia Carlos
  , 2026. Singular volume integral operators associated with constant-coefficient partial differential operators extend the applicability of potential theory to inhomogeneous problems, for example arising from nonlinearities or variable coefficients. Typically the PDE kernels in these operators give rise to singularities at all O(1/h^3 ) volume discretization/evaluation points in a mesh of characteristic size h, while the slowly-decaying nature of such kernels give rise to long-range interactions that require coupling to fast summation algorithms. The presented method uses Green's identities to regularize a wide variety of both scalar-valued and vector-valued volume integral operators by use of a certain regularizing volume density interpolant. The analysis shows how the regularizing effect of the interpolant is global in the sense that the interpolation quality increases in an exactly compensatory fashion as the distance to the Green's function singularity decreases. High-order convergence estimates with tabulated simplex quadratures are established, including with exact representation of curved domains.
• Trapped modes in electromagnetic waveguides
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Chesnel Lucas
  • Fliss Sonia
  Integral Equations and Operator Theory, Springer Verlag, 2026. We consider the Maxwell's equations with perfect electric conductor boundary conditions in three-dimensional unbounded domains which are the union of a bounded resonator and one or several semi-infinite waveguides. We are interested in the existence of electromagnetic trapped modes, i.e. $L^2$ solutions of the problem without source term. These trapped modes are associated to eigenvalues of the Maxwell's operator, that can be either below the essential spectrum or embedded in it. First for homogeneous waveguides, we present different families of geometries for which we can prove the existence of eigenvalues. Then we exhibit certain non homogeneous waveguides with local perturbations of the dielectric constants that support trapped modes. Let us mention that some of the mechanisms we propose are very specific to Maxwell's equations and have no equivalent for the scalar Dirichlet or Neumann Laplacians.
• On some coupled local and nonlocal diffusion models
  
  • Borthagaray Juan Pablo
  • Ciarlet Patrick
  Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2026. We study problems in which a local model is coupled with a nonlocal one. We propose two energies: both of them are based on the same classical weighted $H^1$-semi norm to model the local part, while two different weighted $H^s$-semi norms, with $s \in (0, 1)$, are used to model the nonlocal part. The corresponding strong formulations are derived. In doing so, one needs to develop some technical tools, such as suitable integration by parts formulas for operators with variable diffusivity, and one also needs to study the mapping properties of the Neumann operators that arise. In contrast to problems coupling purely local models, in which one requires transmission conditions on the interface between the subdomains, the presence of a nonlocal operator may give rise to nonlocal fluxes. These nonlocal fluxes may enter the problem as a source term, thereby changing its structure. Finally, we focus on a specific problem, that we consider most relevant, and study regularity of solutions and finite element discretizations. We provide numerical experiments to illustrate the most salient features of the models. (10.1142/S0218202526500442)
  DOI : 10.1142/S0218202526500442
• Waves within a network of slowly time-modulated interfaces: time-dependent effective properties, reciprocity and high-order dispersion
  
  • Darche Michaël
  • Lombard Bruno
  • Touboul Marie
  • Assier Raphaël
  • Guenneau Sebastien
  Proceedings of the Royal Society of London. Series A, Mathematical and physical sciences, Royal Society, The, 2026. We consider wave propagation through a 1D periodic network of slowly time-modulated interfaces. Each interface is modelled by time-dependent spring-mass jump conditions, where mass and rigidity interface parameters are modulated in time. Low-frequency homogenisation yields a leading-order model described by an effective time-dependent wave equation, i.e. a wave equation with effective mass density and Young's modulus which are homogeneous in space but depend on time. This means that time-dependent bulk effective properties can be created by an array where only interfaces are modulated in time. The occurrence of k-gaps in case of a periodic modulation is also analysed. Second-order homogenisation is then performed and leads to an effective model which is reciprocal but encapsulates higher-order dispersive effects. These findings and the limitations of the models are illustrated through time-domain simulations.
• Slip optimization on arbitrary 3D microswimmers: a reduced-dimension and boundary-integral framework
  
  • Bonnet Marc
  • Das Kausik
  • Veerapaneni Shravan
  • Zhu Hai
  SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2026. This article presents a computational framework for determining the optimal slip velocity of a microswimmer with arbitrary three-dimensional geometry suspended in a viscous fluid. The objective is to minimize the hydrodynamic power dissipation required to maintain unit speed along the net swimming direction. By exploiting the linearity of the Stokes equations and the Lorentz reciprocal theorem, we derive an explicit linear operator that maps the tangential surface slip velocity to the resulting rigid-body translational and rotational velocities, effectively decoupling the hydrodynamic boundary value problem from the optimization loop. The a priori infinite-dimensional search space for the slip optimization is reduced to the finite dimension $r$ of rigid-body motions by finding an appropriate subspace of the operator's domain. This reduces the PDE-constrained optimization to a low-dimensional programming problem that can be solved at negligible computational cost once the system matrices are assembled. The optimization algorithm requires 2$r$ auxiliary flow problems that are solved numerically using a high-order boundary integral method. We validate the accuracy of the proposed method and present optimal slip profiles and swimming trajectories for a variety of microswimmer shapes. We investigate the effect of some common geometrical symmetries of the swimmer shape on the resulting optimal motion, and in particular present a modified version of the slip optimization algorithm for axisymmetric shapes, where tangential rigid-body velocities may occur.
• Predicting topologically protected interface state with high-frequency homogenization
  
  • Touboul Marie
  • Lombard Bruno
  • Coutant Antonin
  Comptes Rendus. Mécanique, Académie des sciences (Paris), 2026, 354, pp.269-291. When two semi-infinite periodic media are joined together, a localized interface mode may exist, whose frequency belongs to their common band gap. Moreover, if certain spatial symmetries are satisfied, this mode is topologically protected and thus is robust to defects. A method has recently been proposed to identify the existence and the frequency of this mode, based on the computation of surface impedances at all the frequencies in the gap. In this work, we approximate the surface impedances thanks to highfrequency effective models, and therefore get a prediction of topologically protected interface states while only computing the solution of an eigenvalue problem at the edges of the bandgaps. We also show that the nearby eigenvalues high-frequency effective models give rise to a better approximation of the surface impedance.
• Crouzeix-Raviart elements on simplicial meshes in $d$ dimensions
  
  • Bohne Nis-Erik
  • Ciarlet Patrick
  • Sauter Stefan
  Foundations of Computational Mathematics, Springer Verlag, 2026. In this paper we introduce Crouzeix-Raviart elements of general polynomial order $k$ and spatial dimension $d\geq2$ for simplicial finite element meshes. We give explicit representations of the non-conforming basis functions and prove that the conforming companion space, i.e., the conforming finite element space of polynomial order $k$ is contained in the Crouzeix-Raviart space. We prove a direct sum decomposition of the Crouzeix-Raviart space into (a subspace of) the conforming companion space and the span of the non-conforming basis functions. Degrees of freedom are introduced which are bidual to the basis functions and give rise to the definition of a local approximation/interpolation operator. In two dimensions or for $k=1$, these freedoms can be split into simplex and $(d-1)$ dimensional facet integrals in such a way that, in a basis representation of Crouzeix-Raviart functions, all coefficients which belong to basis functions related to lower-dimensional faces in the mesh are determined by these facet integrals. It will also be shown that such a set of degrees of freedom does not exist in higher space dimension and $k&gt;1$.
• Solving numerically the two-dimensional time harmonic Maxwell problem with sign-changing coefficients
  
  • Chaaban Farah
  • Ciarlet Patrick
  • Rihani Mahran
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2026. We are investigating the numerical solution to the 2D time-harmonic Maxwell equations in the presence of a classical medium and a metamaterial, that is with sign-changing coefficients. As soon as the problem has a (unique) solution, we are able to build a converging numerical approximation based on the finite element method, for which there is no constraint on the meshes related to the sign-changing behavior. To that aim, we use Lagrange finite elements to approximate the scalar potentials appearing in the Helmholtz decomposition of the vector-valued electromagnetic fields. Convergence in strong norm is proven for the fields. Numerical examples illustrate the theory.
• Multiscale methods for wave propagation in materials with sign-changing coefficients
  
  • Chung Eric T.
  • Ciarlet Patrick
  • Jin Xingguang
  • Ye Changqing
  Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal, Society for Industrial and Applied Mathematics, 2026. From a mathematical perspective, the extraordinary properties of metamaterials are often reflected in the coefficients of the governing partial differential equations (PDEs). These coefficients may fall outside the assumptions of classical theory, particularly when the effective dielectric permittivity and/or magnetic permeability are negative. This situation can transform a coercive operator into a non-coercive one, potentially leading to ill-posedness. In this paper, we utilize the Constraint Energy Minimizing Generalized Multiscale Finite Element Method (CEM-GMsFEM), specifically designed for time-harmonic electromagnetic wave problems, where the construction of auxiliary spaces in the original CEM-GMsFEM is tailored to accommodate the sign-changing setting. Based on the framework of T-coercivity theory and resolution conditions, we establish the inf-sup stability and provide an a priori error estimate for the proposed method. The numerical results demonstrate the effectiveness and robustness of our approach in handling such sophisticated coefficient profiles.
• Well-posed homogenized strain-gradient models for linear elastodynamics and elastostatics in arbitrary periodic media
  
  • Cornaggia Rémi
  • Bonnet Marc
  • Rosi Giuseppe
  • El Ouafa Saad
  • Auffray Nicolas
  , 2026. This work develops well-posed homogenized strain-gradient models for linear elastostatics and elastodynamics in periodic media, with a primary focus on elastic wave propagation. Using the classical two-scale asymptotic expansion method, we carry out second-order periodic homogenization for media in $\Rbb^d$ ($d = 2, 3$), with no restriction on the periodicity cell geometry or material distribution. Reciprocity identities applied to suitably chosen pairs of cell solutions provide alternative expressions for the effective stiffness and inertia tensors arising at the leading, first and second orders, substantially reducing the number of cell problems that must actually be solved. Since direct two-scale homogenization beyond leading order generically yields ill-posed effective operators, a Boussinesq-trick procedure is introduced, involving a tunable scalar weight, to recast the resulting fourth-order partial differential equation as a valid strain-gradient elasticity (SGE) model possessing the requisite symmetry, sign-definiteness and coercivity properties. These properties are then used, via the Hille-Yosida theorem, to establish the well-posedness of the corresponding transient initial-value and forced-response problems. Several practically relevant special cases are examined, including centrosymmetric cells, homogeneous mass density and homogeneous elasticity, each yielding simplified model structures. Numerical illustrations on three two-dimensional periodicity cells (square, hexagonal and a non-centrosymmetric chiral lattice) compare the resulting dispersion relations against reference Floquet-Bloch computations and assess transient wave propagation, demonstrating the model's capacity to capture anisotropic and dispersive effects beyond classical elasticity while preserving mathematical well-posedness.
• Spectral theory for periodic operators
  
  • Fliss Sonia
  , 2026, 67. These lecture notes present the main results on the spectral theory of periodic operators with or without perturbations. After recalling the Floquet Bloch transform and the main results of spectral theory, we prove that periodic operators have a band spectrum with possibly band gaps. We will focus in particular on the so-called SSH discrete operator and 1D differential operators for which explicit computations can be done. We will then show few techniques to create gaps by playing with the coefficients. Finally, we will study the effect of perturbations of the periodic coefficients on the spectrum of the underlying operator. In particular, when the perturbation is localized, some discrete spectrum can appear. These lecture was given during the Research School "Théorie Spectrale, Contrôle et Problèmes inverses" in the framework of the CIRM-SMF Program.