Analysis of preconditioning strategies for collocation linear systems

In the past few years several authors have studied the preconditioning of collocation matrices by finite differences (FDs) matrices arising from the associated collocation points. Here we discuss how to solve in an efficient way nonuniform grid FD linear systems, 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. First, we identify the limit spectral distributions of the involved FD-collocation matrix sequences and then we prove that the proposed Toeplitz-based preconditioners assure a clustering at the unity with respect to the eigenvalues in the 1D case. In the 2D case the situation is different so that more appropriate strategies are discussed. A wide numerical experimentation emphasizing the correctness of the theoretical results is also reported.