On accuracy, stability and efficiency of the Newmark method with incomplete solution by multilevel methods

Abstract

The use of an incomplete iterative solver aimed at approximating solution of equilibrium equation resulting from the finite element semidiscretization in space is studied. Our approach is motivated by the fact that: (i) the effective stiffness matrix is well conditioned because of the stabilizing effect of the diagonally dominated mass matrix, (ii) the dominance of the temporal error in implicit computations with large step size, and (iii) the utilization of a class of iterative methods where the primary computational cost is associated with a construction of the preconditioner rather than with an iterative process. One of the primary goals of the present manuscript is to control the temporal and the equilibrium solution errors and to give their a posteriori estimates. Numerical experiments reveal that the computational cost of three cycles conducted with the Generalized Aggregation Multilevel solver [1–4] is comparable to a single back substitution on the source grid, while the resulting equilibrium solution error is negligible compared with the local temporal error. Copyright © 1999 John Wiley & Sons, Ltd.