Non-local domain decomposition methods in electromagnetics – NonLocalDD

This project is oriented toward numerical analysis and scientific computing for large scale simulations of electromagnetic wave propagation in harmonic regime on parallel architectures. It aims at developing new domain decomposition methods (DDM), or improving already existing approaches. Most of the project will focus on non-overlapping strategies. The key novelty of this project is the use of integral operators not only for solutions local to each subdomain, but also for coupling subdomains. 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.

Let us recall that the first step of non-overlapping domain decomposition methods consists 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. Decoupling local problems can be achieved 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, and 3) an iterative algorithm (global solver). This choice has to minimise the overall computational cost, including the cost of local solutions and the number of iterations required for convergence in the iterative process of the global solver.

In a first part of the project, we consider piecewise homogeneous propagation media, and develop DDM fully based on boundary integral operators. In this context, we investigate the potentialities offered by the multi-trace formalism recently developed by the leader of the project together with co-authors. This formalism is extremely modular and adaptable to a wide variety of geometrical and material configurations of the propagation medium. In addition, it provides a proper treatment of junction points (points where three or more subdomains are adjacent), and it is prone to robust preconditioning techniques. Domain decomposition based on this formalism is still unexplored, and deserves to be investigated.

A second part of the project will consider general non-homogeneous propagation media, and construct a new generation of more efficient optimised Schwarz methods i.e. DDM strategies where coupling of subdomains is achieved through the use of Robin type trace operators. In the new strategies we wish to devise, the impedance factors coming into play in Robin traces are non-local boundary integral operators. As strongly suggested by a preliminary work achieved by participants of the project, this new ingredient leads to exponential convergence of global solvers. As such a fast convergence cannot be achieved by already known optimised Schwarz approaches (that feature only algebraic convergence), this would lead to a real breakthrough in domain decomposition for wave propagation problems.

The first two parts of the project are complementary. Hence, the last part of the project is dedicated to domain decomposition strategies taking advantage from both multi-trace formalism for the treatment of junction points, and non-local optimised Schwarz coupling conditions for reaching exponential convergence of global solvers.