Block SSOR preconditionings for high order 3D FE systems (Q582002)

The authors are interested in two-color block preconditionings for variable-order hierarchical finite element discretizations of the Navier equations of structural analysis. For the study reported in the paper, the authors concentrate on domains which can be mapped into square or cubical domains with square or cubical elements. Their goal for variable order p is p-optimality. For this the authors ``require the convergence rate of a p-optimal method to be independent of the approximation order p, the mesh nonuniformity and the coefficients of the differential problem, while the arithmetic costs for one iteration must depend linearly on the number NZ of matrix nonzero entries.'' The requirement of linear growth of arithmetic costs in NZ rules out methods appropriate for fixed p such as incomplete Cholesky preconditioned conjugate gradient method. The authors have performed numerical experiments to study the behavior of several blocking (ordering) methods applied to a selection of model problems. These model problems include Poisson ratios close to 0.5 and irregular meshes. The ordering methods are based on first considering all degrees of freedom associated with a particular geometrical point to be grouped together and ordering the geometric points in strips along one coordinate direction. Within each strip the geometric points are divided into two colors in several different manners and the strips are of varying widths. Detailed numerical results suggest that these orderings result in methods which are nearly p-optimal and the authors indicate which method would be best fo particular classes of problems.