Overview on a selection of recent works in asymptotic analysis for wave propagation problems

We give a brief survey of some recent advances of asymptotic analysis methods applied to wave propagation problem

Essential spectrum of local multi-trace boundary integral operators

Considering pure transmission scattering problems in piecewise constant media, we derive an exact analytic formula for the spectrum of the corresponding local multi-trace boundary integral operators in the case where the geometrical configuration does not involve any junction point and all wave numbers equal. We deduce from this the essential spectrum in the case where wave numbers vary. Numerical evidences of these theoretical results are also presented

Non-local optimized Schwarz method with physical boundaries

We extend the theoretical framework of non-local optimized Schwarz methods as introduced in [Claeys,2021], considering an Helmholtz equation posed in a bounded cavity supplemented with a variety of conditions modeling material boundaries. The problem is reformulated equivalently as an equation posed on the skeleton of a non-overlapping partition of the computational domain, involving an operator of the form "identity + contraction". The analysis covers the possibility of resonance phenomena where the Helmholtz problem is not uniquely solvable. In case of unique solvability, the skeleton formulation is proved coercive, and an explicit bound for the coercivity constant is provided in terms of the inf-sup constant of the primary Helmholtz boundary value problem

Oscillating behaviour of the spectrum for a plasmonic problem in a domain with a rounded corner

We investigate the eigenvalue problem $-\text{div}(\sigma \nabla u) = \lambda u\ (\mathscr{P})$ in a 2D domain $\Omega$ divided into two regions $\Omega_{\pm}$. We are interested in situations where $\sigma$ takes positive values on $\Omega_{+}$ and negative ones on $\Omega_{-}$. Such problems appear in time harmonic electromagnetics in the modeling of plasmonic technologies. In a recent work [15], we highlighted an unusual instability phenomenon for the source term problem associated with $(\mathscr{P})$: for certain configurations, when the interface between the subdomains $\Omega_{\pm}$ presents a rounded corner, the solution may depend critically on the value of the rounding parameter. In the present article, we explain this property studying the eigenvalue problem $(\mathscr{P})$. We provide an asymptotic expansion of the eigenvalues and prove error estimates. We establish an oscillatory behaviour of the eigenvalues as the rounding parameter of the corner tends to zero. We end the paper illustrating this phenomenon with numerical experiments.Comment: Mathematical Modelling and Numerical Analysis (ESAIM: M2AN), 09/12/2016. arXiv admin note: text overlap with arXiv:1304.478

A new variant of the Optimised Schwarz Method for arbitrary non-overlapping subdomain partitions

We consider a scalar wave propagation in harmonic regime modelled by Helmholtz equation with heterogeneous coefficients. Using the Multi-Trace Formalism (MTF), we propose a new variant of the Optimized Schwarz Method (OSM) that can accomodate the presence of cross-points in the subdomain partition. This leads to the derivation of a strongly coercive formulation of our Helmholtz problem posed on the union of all interfaces. The corresponding operator takes the form "identity + contraction"

Derivation, justication and analysis of the Holland method for a model wave propagation problem

The Holland method is a modication of a classical Yee scheme introduced to deal with thin conducting wires when solving Maxwell's equations. This method can be very accurate, but it requires a careful calibration. There still does not exist any systematic recipe for this calibration but, in a previous work we have introduced an augmented Galerkin scheme adapted to the simulation of wave propagation in 2-D domains with small holes and we have shown that, for canonical situations, this numerical scheme could actually provide an automatic process for the calibration of the Holland method. In this talk we would like to describe precisely how, under symmetry assumptions, the traditional Holland scheme used for solving 3-D Maxwell's equations around thin wires actually reduces to this 2-D model situation. We will also present new theoretical results of numerical analysis on this subject

Boundary Element Methods for the Laplace Hypersingular Integral Equation on Multiscreens: a two-level Substructuring Preconditioner

We present a preconditioning method for the linear systems arising from the boundary element discretization of the Laplace hypersingular equation on a $2$-dimensional triangulated surface $\Gamma$ in $\mathbb{R}^3$. We allow $\Gamma$ to belong to a large class of geometries that we call polygonal multiscreens, which can be non-manifold. After introducing a new, simple conforming Galerkin discretization, we analyze a substructuring domain-decomposition preconditioner based on ideas originally developed for the Finite Element Method. The surface $\Gamma$ is subdivided into non-overlapping regions, and the application of the preconditioner is obtained via the solution of the hypersingular equation on each patch, plus a coarse subspace correction. We prove that the condition number of the preconditioned linear system grows poly-logarithmically with $H/h$, the ratio of the coarse mesh and fine mesh size, and our numerical results indicate that this bound is sharp. This domain-decomposition algorithm therefore guarantees significant speedups for iterative solvers, even when a large number of subdomains is used

Nonlocal Optimized Schwarz Methods for time-harmonic electromagnetics

We introduce a new domain decomposition strategy for time harmonic Maxwell's equations that is valid in the case of automatically generated subdomain partitions with possible presence of cross-points. The convergence of the algorithm is guaranteed and we present a complete analysis of the matrix form of the method. The method involves transmission matrices responsible for imposing coupling between subdomains. We discuss the choice of such matrices, their construction and the impact of this choice on the convergence of the domain decomposition algorithm. Numerical results and algorithms are provided

Boundary integral equations of time harmonic wave scattering at complex structures

The first chapter will be a brief recapitulation of well known results concerning layer potentials in the context of wave propagation in harmonic regime. In Chapter 2, we give an overview of the Rumsey reaction principle that is the most popular boundary integral formulation for multi-subdomain scattering, and we present a new alternative integral formulation that seems to be the first boundary integral formulation of the second kind for multi-subdomain scattering in geometrical configurations involving junction points. Chapter 3 is dedicated to the multi-trace formalism which is a completely new approach to boundary integral formulation of multi-subdomain scattering: we briefly describe the local multi-trace formulation, and describe in detail the derivation of the global multi-trace formulation developed by us, as well as its sparsified counterpart that we dubbed quasi-local multi-trace formulation. In Chapter 4 we present a new functional framework adapted to well-posed boundary integral equations for scattering by particular types of object dubbed multi-screens as they take the form of arbitrary arrangements of thin panels of impenetrable material. In Chapter 5 we describe several works on asymptotic modelling in the context wave propagationin harmonic regime. The last chapter presents research perspectives