Modal analysis of the Robin-Schwarz operator (Laurence Halpern, Sorbonne Paris Nord)

12.12.2023 14:00 – 15:00

The Robin-Schwarz operator describes one iteration of the Schwarz domain decomposition algorithm with Robin transmission conditions. Little is known on the spectral properties of this operator which is neither self-adjoint nor compact. We give here for the first time a method to compute the eigenmodes with separation of variables, and prove that they are a complete family. These eigenmodes are of paramount importance for the design and understanding of multi-level domain decomposition methods. The problem inherently couples oscillatory and exponential series, but no result in approximation or spectral theory existing so far can be applied. The analysis calls for multiple tools: regularity results, complex analysis, including non harmonic Fourier series and almost periodic functions. We focus on the special case of 2 by 2 subdomains, but we have little doubts that it opens the door
for more general geometries.

Lieu

Conseil Général 7-9, Room 1-05, Séminaire d'analyse numérique

entrée libre