Section: New Results

Integral equation based domain decomposition

We kept on studying the convergence of classical domain decomposition strategies applied to multi-trace formulations (MTF). In the contribution [18], we present a gentle introduction to multi-trace formalism aimed at the domain decomposition community as well as analytical calculations in simple geometrical configuration where a full analysis of block-Jacoobi applied to MTF is possible. We only consider transmission problems in 1D with one or two interfaces. In [5], we generalize this analysis to arbitrary 2D or 3D transmission problems with arbitrary subdomain partitionning, only assuming that there is no junction point. The analysis holds mainly for completely homogeneous media with no material constrast, and in such a case we determine the spectrum of the multi-trace operator, as well as the spectrum of the Jacobi operator. We show that this spectrum only consists in a finite number of point values. In the more general case where the propagation medium is piecewise constant, this analysis still yields the location of the essential spectrum of the MTF and the Jacobi operator.

This analysis also led to an explicit expression for the inverse of the MTF operators for transmission problems in the case of perfectly homogeneous media. This was studied during the intership of Alan Ayala, and was described and tested numerically in 3D in the proceedings.

The analysis presented in [5] also shows that, in the case of purely homogeneous media, a block Jacobi strategy converges in a number of steps that exactly corresponds to the depth of the adjacency graph of the subdomain partition under consideration, which suggests a close relationship with Optimized Schwarz Methods (OSM), following the ideas of [20]. We investigated this point during the internship of Pierre Marchand, and we exhibited fully explicitely the exact relationship between block-Jacobi-MTF and OSM. Besides, we also generalized the analysis presented in [5] to the case of a competely heterogeneous problem, which involves abstract boundary integral operators that are not easily computable.