Local spectra of adaptive domain decomposition methods

Abstract

We compare the spectra of local generalized eigenvalue problems in different adaptive coarse spaces for overlapping and nonoverlapping domain decomposition methods. In particular, we compare the AGDSW (Adaptive Generalized Dryja--Smith--Widlund), the OS-ACMS (Overlapping Schwarz-Approximate Component Mode Synthesis), and the SHEM (Spectral Harmonically Enriched Multiscale) coarse spaces for overlapping Schwarz methods, the GenEO (Generalized Eigenproblems in the Overlaps) coarse space for FETI-1 and BDD methods, and two approaches based on estimates for the $P_D$ operator for FETI-DP and BDDC methods. Therefore, we consider eight different two-dimensional coefficient functions with jumps ranging from simple channels to a realistic microstructure of a dual-phase steel. We observe significant differences in the width of the gap between good and bad eigenvalues depending on the coefficient distribution. In addition to that, eigenvalue problems involving sophisticated but more expensive harmonic extensions or deluxe-scaling can reduce the number of bad eigenvalues.