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This paper develops an algorithmic scheme to determine initial/starting solutions for iterative linear equation solvers, i.e. preconditioned conjugate gradient, multigrid, SOR and Chebeshev semi-iterative. Employing symbolic operators, multi parameter gauges are introduced to enable individual partition level functional fits of domain decomposed problems. Such gauges are metered by a global minimization of the appropriate energy norm. To remove functional discontinuities on subdomain interfaces, successive applications of a partitioned steepest descent algorithm are employed as a smoothing step. The combination of partitioned initialization and iterative smoothing lead to a robust algorithmic pairing which can provide very accurate starting values for very large simulations. If successively iterated in a series of initialization—steepest descent smoothing cycles, the combined scheme can yield converged results in less iterations than the PCG. This is illustrated by a variety of benchmark examples which de...

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.

The SSOR semi-iterative method is applied to the system of linear equations $A{\bf x}={\bf b}$, where A is an $n \times n$ Hermitian positive definite matrix. We find the bounds for the A-norm of the error vector in terms of the spectral radius of the SSOR iteration matrix.

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.