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Through the introduction of multiple two parameter pairs of scaling factors, a modified formulation of the conjugate gradient scheme is derived. The individual pairs are separately assigned to the various substructure of a model so as to gauge the optimal local search direction and length. During the iteration process, this yields a successions of multiparameter dependent energy states. Seeking the appropriate minima, a vectorized multiparameter formulation of the conjugate gradient scheme is derived. Due to the manner of formulation, the scheme provides both global as well as substructural level orthogonality among successive iterates. To generalize the scheme, both pre and unpreconditioned versions are developed. Overall the methodology provides a potentially significant improvement in computational robustness. This is benchmarked in a series of numerical experiments.

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 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.