Recently an algebraic multilevel incomplete factorization method for solving large linear systems with the Stieltjes matrices has been proposed. This method is a combination of two well-known techniques: algebraic multilevel (AMLI) and incomplete factorization. However, the efficiency of this method strongly depends on the choice of the relaxation parameter θ, an optimal value of which depends on the problem to be solved. In the present paper we study this dependence theoretically and propose a new method, that dynamically computes the corresponding problem–dependence optimal value of θ, and use it to construct an approximation of Schur's complement as a new matrix on the lower level in the AMLI framework.

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