Coarse spaces using extended generalized eigenproblems for heterogeneous Helmholtz problems

An abstract construction of coarse spaces for non-Hermitian problems and non-Hermitian domain decomposition preconditioners based on extended generalized eigenproblems was proposed in [Nataf and Parolin, arXiv:2404.02758] and analyzed on the matrix formulation. Building upon this work, we consider instead here the specific case of heterogeneous Helmholtz problems, and the derivation and analysis is performed at the continuous level. Albeit different from its derivation, its use of oversampling and the underlying eigenproblems, our approach shares similarities with the methods of Hu and Li [SIAM J. Numer. Anal, 63(2), 716-743, 2025] and Ma, Alber, Scheichl and Zhang [J. Sci. Comput., 105(3), No. 99, 2025].