This project in scientic computing aims at developing new domain decomposition methods for massively parallel simulation of electromagnetic waves in harmonic regime. The specificity of the approach that we propose lies in the use of integral operators not only for solutions local to each subdomain, but for coupling subdomains as well. The novelty of this project consists, on the one hand, in exploiting multi-trace formalism for domain decomposition and, on the other hand, considering optimized Schwarz methods relying on Robin type transmission conditions involving quasi-local integral operators.

Scientific and technical goals

For the numerical solution to PDEs, non overlapping domain decomposition methods (DDM) consist first of all in partitioning the computational domain in subdomains. The solution to equations local to each subdomain is then allocated to a local solver (corresponding to a processor), and local problems are coupled by means of transmission conditions. Local problems can be decoupled by an iterative algorithm so that the whole computation is parallelized. From this perspective, the key to an efficient DDM rests on a wise choice of

  1. local solvers,
  2. coupling conditions,
  3. an iterative algorithm (global solver),

so as to minimize the computational cost of local solutions and the number of iterations required for convergence in the global solver algorithm. Such methods have been largely developed for coercive elliptic problems. Wave equations in harmonic regime do not have this property, which is the reason why domain decomposition, in this context, remains a challenge: in complex media whose dimensions are much larger that the wavelength, the simulation of wave propagation is either difficult, or simply out of reach in many applications of interest in the industry, even with the most powerful supercomputers. To address this challenge, we propose two new classes of DDM

  • Methods based on systems of coupled boundary integral equations reformulating wave scat- tering problems in the case of piecewise homogeneous media. Boundary integral formulations are one of the most efficient tools for the numerical simulation of wave scattering, and have met a growing interest in the industry for about two decades, as they substantially lighten the mesh generation effort and drastically reduce the size of linear systems to be solved. For the same number of degrees of freedom, these methods are more precise than classical volumic methods, more robust at high frequency and less dispersive. Domain decomposition in this context represent a rather unexplored field.

  • Methods relying on a quasi-local coupling scheme for the solution of wave scattering in complex media by means of finite elements. In this case, both homogeneous and inhomogeneous media can be taken into account and high order finite elements can be used locally if necessary. Such methods have already been widely developed over the past decades. The specificity in the present proposal is to devise a new approach to reach the geometric convergence of the iterative algorithm (which no other strategy known to us can provide at the moment). The use of integral operators to perform the coupling conditions should make this possible.

The common feature shared by both above mentionned approaches, that is also a point of novelty of this project, is the use of integral operators for achieving coupling between sub-domains. In both cases, our methods will rely on the most recent results on the so-called multi-trace formalism that had been introduced by Claeys, Jerez and Hiptmair [1] as a way to derive coupling schemes between integral operators associated to arbitrarily arranged subdomains. One remarkable strength of this formalism is its proper treatment of junction points i.e. points where three or more subdomains abut (a common but tricky situation in the context of domain decomposition...).

For both research directions i) and ii), we aim at developing a complete work of applied mathematics, including the devise of original algorithms, and the implementation of computational codes based on these algorithms, as well as their mathematical analysis and their discretization. Programming tasks of the present project shall include implementations on parallel architectures so as to demonstrate scalability of the methods under investigation. Following the two points i) and ii) mentioned above, our project is structured according to two research directions for which we now list the points to be tackled.

Research direction 1:
Domain decomposition based on boundary integral operators

In this part of the project we will typically consider a problem of scattering by composite objects containing both dielectric and metallic parts, and we will look for domain decomposition strategies involving boundary integral operators both for local solutions, and for coupling subdomains, taking advantage of the multi-trace formalism. In a first part, we will focus on local solvers, and try to invent or improve boundary integral methods providing accurate, well-conditioned local solvers adapted both to complex geometries and domain decomposition. We will consider cases where scatterers admit the following features:

  • scatterer containing a combination of metallic and dielectric parts;
  • metallic thin sheets (so-called "screens"), possibly admitting several branches;
  • composite scatterers admitting screens at interfaces between dielectric parts;
  • thin layers of dielectric material.

We will analyze each of these situations, trying each time to derive efficient preconditioners. The work associated to this part of the project will be an extension of the existing multi-trace formalism, where the considered solvers will be optimized from a numerical point of view.

Then we will also investigate new global DDM solvers based on multi-trace boundary integral operators. Examining both the local and global variant of multi-trace formulations, we will study the spectrum of iteration operators associated to standard global strategies such as block Jacobi or Gauss-Seidel, and achieve numerical tests on these approaches.

Taking advantage of the strong algebraic properties satised by integral operators, such as Calderón formula, we will also look for new parallel block direct solvers for multi-trace formula- tions. To investigate on the possibility of a scalable method, we propose to derive a coarse space correction strategy inspired by global multi-trace formulations that are fully non-local equations coupling all subdomains.

Finally we will try to use compression techniques, such as hierarchical matrices, in order to speed up the coupling between subdomains. These techniques, developed over the last ten years, in order to bring dense matrix-vector products to an almost linear complexity, are commonly applied to local solvers but, to our knowledge, their use to couple subdomains has never been considered.

Research direction 2:
Non-local optimized Schwarz methods

The pioneering work of Després [3] then Collino, Ghanemi and Joly [2] and Gander, Magoules and Nataf [4] have shown that it is mandatory, in the context of wave equations, to use impedance type transmission conditions in the coupling of subdomains in order to obtain convergence of the DDM. In the approaches considered so far in the literature, the impedance operator involved in the transmission conditions was always local (a scalar in the most simplest cases). These methods lead to algebraic convergence of the DDM in the best cases.

In a recent work, F. Collino, P. Joly and M. Lecouvez (in the context of his PhD. thesis at CEA-Cesta) have observed that using non-local impedances such as integral operators could lead to an exponential convergence of the DDM. One of the strengths of this approach is to rely on a solid theoretical basis that systematically guaranties geometrical convergence, provided that certain properties of injectivity, surjectivity and positivity (in suitable trace spaces) are satised by the impedance. This operator may be constructed by means of fractional pseudo-differential opera- tors involving truncated kernel operators (with Riesz kernel). The developments that we propose hereafter aim at continuing and extending this work.

  • Development of a unified convergence theory admitting the widest possible scope: irregular surfaces, influence of the number of subdomains;
  • Calibrating the parameters involved in the transmission conditions, so as to optimize the convergence rate of the DDM, on the basis of analytical studies of canonical situations;
  • Development of spatial discretization methods, compatible with the continuous theoretical framework, and numerical analysis (uniformity of the h -convergence rate ?);
  • Study of various localisation processes for coupling conditions, and influence of this choice on the convergence of the numerical method;
  • Treatment of multiple junctions: the available theory excludes the presence of multiple junctions that require a particular approach (based on the multi-trace formalism for example);
  • Implementation and validation of numerical methods based on the code Montjoie, and tests on realistic cases, with a particular care dedicated to parallelization aspects.

We will examine these questions, first in the context of Helmholtz equation (that is a relevant model in 2-D electromagnetics), and for Maxwell's equations in 3-D. In this last case, there will appear particular difficulties related to the functional framework specific to the Maxwell system, to the structure of trace spaces, and to more sophisticated properties that should be satisfied by the integral operators involved in the construction of transmission conditions.