Spectral Analysis of a Preconditioned Iterative Method for the Convection-Diffusion Equation

The convergence features of a preconditioned algorithm for the convection-diffusion equation based on its diffusion part are considered. Analyses of the distribution of the eigenvalues of the preconditioned matrix in arbitrary dimensions and of the fundamental parameters of convergence are provided, showing the existence of a proper cluster of eigenvalues. The structure of the cluster is not influenced by the discretization. An upper bound on the condition number of the eigenvector matrix under some assumptions is provided as well. The overall cost of the algorithm is O(n), where n is the size of the underlying matrices.