Residual-Minimizing Krylov Subspace Methods for Stabilized Discretizations of Convection-Diffusion Equations

We discuss the behavior of the minimal residual method applied to stabilized discretizations of oneand two-dimensional model problems for the stationary convection-diffusion equation. In the one-dimensional case, it is shown that eigenvalue information for estimating the convergence rate of the minimal residual method is highly misleading due to the strong nonnormality of these operators for large grid Péclet numbers. It is also shown that the field of values is a more reliable tool for assessing the convergence rate. In the two-dimensional model problems considered, we observe two distinct phases in the convergence of the iterative method: the first determined by the field of values and the second by the spectrum. We conjecture that the first phase lasts as long as the longest streamline takes to traverse the grid with the flow.