The eigenvalue spectrum of domain decomposed preconditioners (Q1181539)

The construction of domain decomposed preconditioners is first described for the solution of linear, second-order elliptic partial differential equations with Dirichlet boundary values and some basic lemmas with respect to the matrices are proved. Then the eigenvalue distribution of the preconditioned operator is analyzed, and it is shown that the eigenspectrum is a perturbation of the union of the spectra of the preconditioned systems on the subdomains and on the interfaces. The general theoretical results are illustrated in more detail for the case of a rectangular domain that is splitted into two strips, that reveals the interplay between the eigendecomposition of the subdomain blocks and the eigendecomposition of the Schur complement blocks. Some derived refined results may give an explanation of the numerical behaviour of the methods as illustrated in examples.