DYNAMICAL MULTILEVEL SCHEMES FOR THE SOLUTION OF EVOLUTION-EQUATIONS BY HIERARCHICAL FINITE-ELEMENT DISCRETIZATION

The full numerical simulation of turbulent hows in the context of indu strial applications remains a very challenging problem. One of the dif ficulties is the presence of a large number of interacting scales of d ifferent orders of magnitude ranging from the macroscopic scale to the Kolmogorov dissipation scale. In order to better understand and simul ate these interactions, new algorithms of incremental type have been r ecently introduced, stemming from the theory of infinite dimensional d ynamical systems, see, e.g., the algorithms of Foias, Jolly et al. (19 88), Foias, Manley and Temam (1988), Laminie et al. (1993, 1994), Mari on and Temam (1989, 1990), Temam (1990). These algorithms are based on decompositions of the unknown functions into a large and a small scal e component, one of the underlying ideas being to approximate the corr esponding attractor. In the context of finite elements methods, the de composition of solution into small and large scale components appears when we consider hierarchical bases (Yserentant, 1986; Zienkiewicz et al., 1982). In the present article we derive several estimates of the size of the structures for the linear and nonlinear terms which corres pond to interactions of different hierarchical components of the veloc ity held, and also their time variation. The one-step time variation o f these terms can be smaller than the expected accuracy of the computa tion. Using this remark, we implement an adaptive spatial and temporal multilevel algorithm which treats differently the small and large sca le components of the flow. We derive several a priori estimates in ord er to study the perturbation introduced into the approximated equation s. All the interaction terms between small and large structures are fr ozen during several time steps. Finally we implement the multilevel me thod in order to simulate a bidimensional flow described by the Burger s' equations. We perform a parametric study of the procedure and its e fficiency. The gain on CPU time is significant and the trajectories co mputed by our multi-scale method remain close to the trajectories obta ined with the classical Galerkin method. (C) 1997 Elsevier Science B.V .