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

Stable approximation of Helmholtz solutions by evanescent plane waves

Solutions of the Helmholtz equation are known to be well approximated by superpositions of propagative plane waves. This observation is the foundation of successful Trefftz methods. However, when too many plane waves are used, the computation of the expansion is known to be numerically unstable. We prove that exponentially large coefficients in the expansion are unavoidable and we explain how this causes instability. In this work, we show that the Helmholtz solutions on a disk can be exactly represented by a continuous superposition of evanescent plane waves, generalizing the standard Herglotz representation. Here, by evanescent plane waves, we mean exponential plane waves with complex-valued propagation vector, whose absolute value decays exponentially in one direction. In addition, the density in this representation is proved to be uniformly bounded in a suitable weighted Lebesgue norm, hence overcoming the instability observed with propagative plane waves and paving the way for stable discrete expansions. In view of practical implementations, discretization strategies are investigated. We construct suitable finite-dimensional sets of evanescent plane waves using sampling strategies in a parametric domain. Provided one uses sufficient oversampling and regularization, numerical experiments show that the resulting approximations are both controllably accurate and numerically stable

Non-local Impedance Operator for Non-overlapping DDM for the Helmholtz Equation

In the context of time harmonic wave equations, the pioneering work of B. Després [4] has shown that it is mandatory to use impedance type transmission conditions in the coupling of sub-domains in order to obtain convergence of nonoverlapping domain decomposition methods (DDM). In later works [2, 3], it was observed that using non-local impedance operators leads to geometric convergence, a property which is unattainable with local operators. This result was recently extended to arbitrary geometric partitions, including configurations with cross-points, with provably uniform stability with respect to the discretization parameter [1]. We present a novel strategy to construct suitable non-local impedance operators that satisfy the theoretical requirements of [1] or [2, 3]. It is based on the solution of elliptic auxiliary problems posed in the vicinity of the transmission interfaces. The definition of the operators is generic, with simple adaptations to the acoustic or electromagnetic settings, even in the case of heterogeneous media. Besides, no complicated tuning of parameters is required to get efficiency. The implementation in practice is straightforward and applicable to sub-domains of arbitrary geometry, including ones with rough boundaries generated by automatic graph partitioners. We first provide in Section 1 a general definition of this novel transmission operator in a two-domain configuration. In Section 2 we then study more quantitatively the convergence in the geometric configuration of a closed wave-guide. Section 3 illustrates the results using actual finite element computations

Non overlapping Domain Decomposition Methods for Time Harmonic Wave Problems

The domain decomposition method (DDM) initially designed, with the celebrated paper of Schwarz in 1870 as a theoretical tool for partial differential equations (PDEs) has become, since the advent of the computer and parallel computing techniques, a major tool for the numerical solution of such PDEs, especially for large scale problems. Time harmonic wave problems offer a large spectrum of applications in various domains (acoustics, electromagnetics, geophysics, ...) and occupy a place of their own, that shines for instance through the existence of a natural (possibly small) length scale for the solutions: the wavelength. Numerical DDMs were first invented for elliptic type equations (e.g. the Laplace equation), and even though the governing equations of wave problems (e.g. the Helmholtz equation) look similar, standard approaches do not work in general

Robust treatment of cross points in Optimized Schwarz Methods

In the field of Domain Decomposition (DD), Optimized Schwarz Method (OSM) appears to be one of the prominent techniques to solve large scale time-harmonic wave propagation problems. It is based on appropriate transmission conditions using carefully designed impedance operators to exchange information between sub-domains. The efficiency of such methods is however hindered by the presence of cross-points, where more than two sub-domains abut, if no appropriate treatment is provided. In this work, we propose a new treatment of the cross-point issue for the Helmholtz equation that remains valid in any geometrical interface configuration. We exploit the multi-trace formalism to define a new exchange operator with suitable continuity and isometry properties. We then develop a complete theoretical framework that generalizes classical OSM to partitions with cross points and contains a rigorous proof of geometric convergence, uniform with respect to the mesh discretization, for appropriate positive impedance operators. Extensive numerical results in 2D and 3D are provided as an illustration of the performance of the proposed method

Stable approximation of Helmholtz solutions in the disk by evanescent plane waves

Superpositions of plane waves are known to approximate well the solutions of the Helmholtz equation. Their use in discretizations is typical of Trefftz methods for Helmholtz problems, aiming to achieve high accuracy with a small number of degrees of freedom. However, Trefftz methods lead to ill-conditioned linear systems, and it is often impossible to obtain the desired accuracy in floating-point arithmetic. In this paper we show that a judicious choice of plane waves can ensure high-accuracy solutions in a numerically stable way, in spite of having to solve such ill-conditioned systems. Numerical accuracy of plane wave methods is linked not only to the approximation space, but also to the size of the coefficients in the plane wave expansion. We show that the use of plane waves can lead to exponentially large coefficients, regardless of the orientations and the number of plane waves, and this causes numerical instability. We prove that all Helmholtz fields are continuous superposition of evanescent plane waves, i.e., plane waves with complex propagation vectors associated with exponential decay, and show that this leads to bounded representations. We provide a constructive scheme to select a set of real and complex-valued propagation vectors numerically. This results in an explicit selection of plane waves and an associated Trefftz method that achieves accuracy and stability. The theoretical analysis is provided for a two-dimensional domain with circular shape. However, the principles are general and we conclude the paper with a numerical experiment demonstrating practical applicability also for polygonal domains

FEM-BEM Coupling for Electromagnetism with the Sparse Cardinal Sine Decomposition

This paper presents a FEM-BEM coupling method suitable for the numerical simulation of the electromagnetic scattering of objects composed of dielectric materials and perfect electric conduc- tors. The originality of the approach lies in part in the use of the newly proposed Sparse Cardinal Sine Decomposition SCSD) method for the BEM part of the computation and the fact that the simulation software is almost entirely written in MATLAB. The performance of the method is illustrated by the computation of the electromagnetic scattering by an UAV-like object with two RAM regions proposed in the workshop ISAE EM 2016

FEM-BEM Coupling for Electromagnetism with the Sparse Cardinal Sine Decomposition★,★★

This paper presents a FEM-BEM coupling method suitable for the numerical simulation of the electromagnetic scattering of objects composed of dielectric materials and perfect electric conduc- tors. The originality of the approach lies in part in the use of the newly proposed Sparse Cardinal Sine Decomposition SCSD) method for the BEM part of the computation and the fact that the simulation software is almost entirely written in MATLAB. The performance of the method is illustrated by the computation of the electromagnetic scattering by an UAV-like object with two RAM regions proposed in the workshop ISAE EM 2016