This paper provides a proof of robustness of the restricted additive Schwarz preconditioner with harmonic overlap (RASHO) for the second order elliptic problems with jump coefficients. By analyzing the eigenvalue distribution of the RASHO preconditioner, we prove that the convergence rate of preconditioned conjugate gradient method with RASHO preconditioner is uniform with respect to the large jump and meshsize.

In this paper a feasible direction method is presented to find all efficient extreme points for a special class of multiple objective linear fractional programming problems, when all denominators are equal. This method is based on the conjugate gradient projection method, so that we start with a feasible point and then a sequence of feasible directions towards all efficient adjacent extremes of the problem can be generated. Since methods based on vertex information may encounter difficulties as the problem size increases, we expect that this method will be less sensitive to problem size. A simple production example is given to illustrate this method.

In this paper, we propose a new two-level preconditioned C-G method which uses the quadratic smoothing and the linear correction in distorted but topo-logically structured grid. The CPU time of this method is less than that of the multigrid preconditioned C-G method (MGCG) using the quadratic element, but their accuracy is almost the same. Numerical experiments and eigenvalue analysis are given and the results show that the proposed two-level preconditioned method is efficient.