Amplitude-based Generalized Plane Waves: new Quasi-Trefftz functions for scalar equations in 2D

Generalized Plane Waves (GPWs) were introduced to take advantage of Trefftz methods for problems modeled by variable coefficient equations. Despite the fact that GPWs do not satisfy the Trefftz property, i.e. they are not exact solutions to the governing equation, they instead satisfy a quasi-Trefftz property: they are only approximate solutions. They lead to high order numerical methods, and this quasi-Trefftz property is critical for their numerical analysis. The present work introduces a new family of GPWs, amplitude-based. The motivation lies in the poor behavior of the phase-based GPW approximations in the pre-asymptotic regime, which will be tamed by avoiding high degree polynomials within an exponential. The new ansatz is introduces higher order terms in the amplitude rather than the phase of a plane wave as was initially proposed. The new functions' construction and the study of their interpolation properties are guided by the roadmap proposed in [16]. For the sake of clarity, the first focus is on the two-dimensional Helmholtz equation with spatially-varying wavenumber. The extension to a range of operators allowing for anisotropy in the first and second order terms follows. Numerical simulations illustrate the theoretical study of the new quasi-Trefftz functions

Fast, adaptive, high order accurate discretization of the Lippmann-Schwinger equation in two dimension

We present a fast direct solver for two dimensional scattering problems, where an incident wave impinges on a penetrable medium with compact support. We represent the scattered field using a volume potential whose kernel is the outgoing Green's function for the exterior domain. Inserting this representation into the governing partial differential equation, we obtain an integral equation of the Lippmann-Schwinger type. The principal contribution here is the development of an automatically adaptive, high-order accurate discretization based on a quad tree data structure which provides rapid access to arbitrary elements of the discretized system matrix. This permits the straightforward application of state-of-the-art algorithms for constructing compressed versions of the solution operator. These solvers typically require $O(N^{3/2})$ work, where $N$ denotes the number of degrees of freedom. We demonstrate the performance of the method for a variety of problems in both the low and high frequency regimes.Comment: 18 page

Integral Equation Methods for Electrostatics, Acoustics, and Electromagnetics in Smoothly Varying, Anisotropic Media

[EN] We present a collection of well-conditioned integral equation methods for the solution of electrostatic, acoustic, or electromagnetic scattering problems involving anisotropic, inhomogeneous media. In the electromagnetic case, our approach involves a minor modification of a classical formulation. In the electrostatic or acoustic setting, we introduce a new vector partial differential equation, from which the desired solution is easily obtained. It is the vector equation for which we derive a well-conditioned integral equation. In addition to providing a unified framework for these solvers, we illustrate their performance using iterative solution methods coupled with the FFT-based technique of [F. Vico, L. Greengard, M. Ferrando, J. Comput. Phys., 323 (2016), pp. 191-203] to discretize and apply the relevant integral operators.The work of the authors was partially supported by the Spanish Ministry of Science and Innovation under project TEC2016-78028-C3-3-P and the U.S. Department of Energy under grant DE-FG02-86ER53223.Imbert-Gérard, L.; Vico Bondía, F.; Greengard, L.; Ferrando Bataller, M. (2019). Integral Equation Methods for Electrostatics, Acoustics, and Electromagnetics in Smoothly Varying, Anisotropic Media. SIAM Journal on Numerical Analysis. 57(3):1020-1035. https://doi.org/10.1137/18M1187039S1020103557

Analyse mathématique et numérique de problèmes d'ondes apparaissant dans les plasmas magnétiques

This dissertation investigates mathematical and numerical aspects of some wave phenomena appearing in magnetic plasmas. Inorder to model a probing technique for fusion plasmas, called reflectometry, a particular form of Maxwell s equations is studied. Inthe model, the dielectric tensor presents vanishing eigenvalues and diagonal terms. The study of the dispersion relation evidencestwo kinds of phenomena: cut-offs and resonances if the wave number goes either to zero or to infinity.Part I of the thesis gathers the theoretical results. The main novelty consists in the definition of a resonant solution. Indeed, becauseof a smooth vanishing sign-changing coefficient, the solution may be singular: one of its components may be non-integrable. However,using a limit absorption principle, a resonant solution is explicitly obtained by studying the integrable solutions of the regularizedsystem plus a limiting process. The theoretical expression of the singularity is validated by numerical tests concerning the regularizedsystem as the regularizing term goes to zero.Part II focuses on the numerical results. It includes the design of a new numerical method adapted to smooth coefficients. Themethod is based on the Ultra Weak Variational Formulation but requires specific shape functions, designed as local approximationsof the adjoint equation. The convergence analysis of the method is performed in one dimension, for two dimensions the designprocedure and the interpolation property of the shapes functions are detailed. The resulting high order method numerically tacklesthe approximation of cut-offs while the approximation of resonant solutions is still very challenging.Cette thèse étudie les aspects mathématiques et numériques de phénomènes d ondes dans les plasmas magnétiques. La réflectométrie,une technique de sonde des plasmas de fusion, est modélisée par les équations de Maxwell. Le tenseur de permittivité présentedans ce model des valeurs propres ainsi que des termes diagonaux qui s annulent. La relation de dispersion met en évidence deuxphénomènes cruciaux : coupures et résonances, lorsque le nombre d onde s annule ou tend vers l infini.La partie I rassemble les résultats numériques. La grande nouveauté réside dans la définition d une solution résonante. En effet, àcause des coefficients s annulant continuement en changeant de signe, la solution peut être singulière, i.e. avoir une composante nonintégrable. Cependant, grâce au principe d absorption limite, une solution résonante est explicitement définie comme la limite desolutions intégrables du problème régularisé. L expression théorique de la singularité est validée par des tests numériques du passageà la limite.La partie II concerne l approximation numérique. Elle comprend la mise en place d une nouvelle méthode numérique adaptée auxcoefficients réguliers. Celle-ci est basée sur la Formulation Variationnelle Ultra Faible mais nécessite des fonctions de base spécifiques,construites comme approximations locales du problème adjoint. L analyse de convergence est effectuée en dimesion un, en dimensiondeux la construction des fonctions de base et leur propritété d interpolation sont détaillées. La méthode d ordre élevé obtenue permetde simuler le phénomène de coupure tandis que simuler le phénomène de résonance en dimension deux reste un défi.PARIS-BIUSJ-Mathématiques rech (751052111) / SudocSudocFranceF