65N22 Solution of discretized equations [See also 65Fxx, 65Hxx]
We introduce and analyze nonsmooth Schur-Newton methods for a class of nonsmooth saddle point problems. The method is able to solve problems where the primal energy decomposes into a convex smooth part and a convex separable but nonsmooth part. The method is based on nonsmooth Newton techniques for an equivalent unconstrained dual problem. Using this we show that it is globally convergent even for inexact evaluation of the linear subproblems.
Analysis of the second phase of the GMRES convergence for a convection-diffusion model problem
(2012)
It is well konwn that GMRES applied to linear algebraic systems arising from a convection-diffusion model problem that has been discretized by the streamline upwind Petrov-Galerkin (SUPG) method, typically displays two distinct phases of convergence: a slow initial phase followed by a convergence acceleration in the second phase. This paper complements the known results on the length of the initial phase by analyzing how the acceleration in the second phase of convergence is related to the mesh Peclet number and the choice of the stabilization parameter in the SUPG discretization. The analysis is based on some new expressions and bounds for the GMRES residuals, which can be of general interest.
Here we analyze algebraic multilevel methods applied to non-symmetric M-matrices.
We consider two types of multilevel approximate block factorizations.
The first one is related to the AMLI method.
The second method
is the multiplicative counterpart of the AMLI approach which we call
multiplicative algebraic multilevel method, the MAMLI method. The MAMLI method is closely related to certain geometric
and algebraic multigrid methods like the AMGr method. Although these
multilevel methods work very well
in practice for many problems, there is not that much known about
theoretical convergence
properties for non-symmetric problems.
Here, we establish
convergence results and comparison results between AMLI and MAMLI
multilevel methods
applied to non-symmetric M-matrices.
We establish theoretical comparison results for algebraic multi-level
methods applied
to nonsingular non-symmetric M-matrices.
We consider two types of multi-level approximate block factorizations or AMG methods, the
AMLI and the MAMLI method.
We compare
the spectral radii
of the iteration matrices of these methods. This comparison
shows, that the spectral radius of the MAMLI method is less than or equal to
the spectral radius of the AMLI method.
Moreover, we establish how the quality of the approximations
in the block factorization effects the spectral radii of the
iteration
matrices. We prove comparisons results
for different approximation of the fine grid block as well as for the
used Schur
complement. We also establish a theoretical comparison between the
AMG methods and the classical block Jacobi and block Gauss-Seidel methods.