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We obtain conditions for the convergence of SSOR and USSOR methods when the matrix A has a special form (so-called “red-black” ordering), using some vectorial norms. We give characterizations for the H-matrix with respect to the USSOR iteration matrix. Our results extend the work of Alefeld and Krishna.

To improve the numerical stability of nonlinear equation solvers, a partitioned multiply scaled constraint scheme is developed. This scheme enables hierarchical levels of control for nonlinear equation solvers. To complement the procedure, partitioned convergence checks are established along with self-adaptive partitioning schemes. Overall, such procedures greatly enhance the numerical stability of the original solvers. To demonstrate and motivate the development of the scheme, the problem of nonlinear heat conduction is considered. In this context the main emphasis is given to successive substitution-type schemes. To verify the improved numerical characteristics associated with partitioned multiply scaled solvers, results are presented for several benchmark examples.

To improve the numerical stability of nonlinear equation solvers, a partitioned multiply scaled constraint scheme is developed. This scheme enables hierarchical levels of control for nonlinear equation solvers. To complement the procedure, partitioned convergence checks are established along with self-adaptive partitioning schemes. Overall, such procedures greatly enhance the numerical stability of the original solvers. To demonstrate and motivate the development of the scheme, the problem of nonlinear heat conduction is considered. In this context the main emphasis is given to successive substitution-type schemes. To verify the improved numerical characteristics associated with partitioned multiply scaled solvers, results are presented for several benchmark examples.

The Unsymmetric Successive Overrelaxation (USSOR) iterative method is applied to the solution of the system of linear equationsA x=b, whereA is annxn nonsingular matrix. We find the values of the relaxation parameters ? and $$\bar \omega $$ for which the USSOR iterative method converges. Then we characterize those matrices which are equimodular toA and for which the USSOR iterative method converges.

The shortcomings associated with successive substitution nonlinear solvers are circumvented by introducing a so-called block constraint methodology. Specifically, by partitioning the governing field equations into various subsets, constraints can be applied individually to each separate block. To illustrate the formal properties of the scheme, a series of lemmas and a theorem will be developed. These define its convergence properties. Due to the generality of the approach taken, virtually any form of functional constraint may be employed.