Using Local Spectral Information in Domain Decomposition Methods – A Brief Overview in a Nutshell

Abstract

AbstractFor second‐order elliptic partial differential equations large discontinuities in the coefficients yield ill‐conditioned stiffness matrices. The convergence of domain decomposition methods (DDM) can be improved by incorporating (numerically computed) local eigenvectors into the coarse space. Different adaptive coarse spaces for DDM have been constructed and used successfully. For two‐level Schwarz, FETI‐1 and BDD methods, adaptive coarse spaces with a rigorous theoretical basis are known for 2D and 3D. Although successfully in use for almost a decade, a theory for adaptive coarse spaces for FETI‐DP and BDDC was lacking. While the problem was recently settled for 2D, the estimate for the 3D adaptive algorithm required improved coarse spaces. We give an brief overview of the literature, i. e., the different known approaches, and show numerical results for a specific adaptive FETI‐DP method in 3D, where the condition number bound could only recently be proven. (© 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)