GMRES convergence analysis for a convection-diffusion model problem

When GMRES [Y. Saad and M. H. Schultz, SIAM J. Sci. Statist. Comput.}, 7 (1986), pp. 856--869] is applied to streamline upwind Petrov--Galerkin (SUPG) discretized convection-diffusion problems, it typically exhibits an initial period of slow convergence followed by a faster decrease of the residual norm. Several approaches were made to understand this behavior. However, the existing analyses are solely based on the matrix of the discretized system and they do not take into account any influence of the right-hand side (determined by the boundary conditions and/or source term in the PDE). Therefore they cannot explain the length of the initial period of slow convergence which is right-hand side dependent. We concentrate on a frequently used model problem with Dirichlet boundary conditions and with a constant velocity field parallel to one of the axes. Instead of the eigendecomposition of the system matrix, which is ill conditioned, we use its orthogonal transformation into a block-diagonal matrix with nonsymmetric tridiagonal Toeplitz blocks and offer an explanation of GMRES convergence. We show how the initial period of slow convergence is related to the boundary conditions and address the question why the convergence in the second stage accelerates

On optimal short recurrences for generating orthogonal Krylov subspace bases

We analyze necessary and sufficient conditions on a nonsingular matrix A such that, for any initial vector $r_0$, an orthogonal basis of the Krylov subspaces ${\cal K}_n(A,r_0)$ is generated by a short recurrence. Orthogonality here is meant with respect to some unspecified positive definite inner product. This question is closely related to the question of existence of optimal Krylov subspace solvers for linear algebraic systems, where optimal means the smallest possible error in the norm induced by the given inner product. The conditions on A we deal with were first derived and characterized more than 20 years ago by Faber and Manteuffel (SIAM J. Numer. Anal., 21 (1984), pp. 352–362). Their main theorem is often quoted and appears to be widely known. Its details and underlying concepts, however, are quite intricate, with some subtleties not covered in the literature we are aware of. Our paper aims to present and clarify the existing important results in the context of the Faber–Manteuffel theorem. Furthermore, we review attempts to find an easier proof of the theorem and explain what remains to be done in order to complete that task