M. Jung and S. V. Nepomnyaschikh
Variable preconditioning procedures for elliptic problems
Preprint SFB393/96-22, Sonderforschungsbereich 393, TU Chemnitz-Zwickau
For solving systems of grid equations approximating elliptic boundary value problems a method of constructing variable preconditioning procedures is presented. The main purpose is to discuss how an efficient preconditioning iterative procedure can be constructed in the case of elliptic problems with disproportional coefficients, e.g. equations with a large coefficient in the reaction term (or a small diffusion coefficient). The optimality of the suggested technique is based on fictitious space and multilevel decomposition methods. Using an additive form of the preconditioners, we introduce factors into the preconditioners to optimize the corresponding convergence rate. The optimization with respect to these factors is used at each step of the iterative process. The application of this technique to two-level $p$-hierarchical preconditioners and domain decomposition methods is considered too.