Analysis of preconditioning strategies for collocation linear systems (Q1399920)

The authors discuss some possible efficient ways to solve nonuniform linear systems arising from discretization by finite differences (FD) of elliptic operators, including those related to a generic FD-collocation preconditioner. The main idea is based on a further step of preconditioning defined in terms of diagonal and Toeplitz matrices. The limit spectral distributions of the involved FD-collocation matrix sequences is identified and then authors proved that the proposed Toeplitz-based preconditioners assure a clustering at the unity with respect to the eigenvalues in the one-dimensional case. The two-dimensional case is discussed. The spectral distribution of the resulting sequences of FD matrices is determined and the clustering properties of the Toeplitz+diagonal preconditioning is studied. Some numerical experiments emphasizing the correctness of the theoretical results.