Divergence Stability and Adaptive Dimensional Reduction

Prof. Soren Jensen

Department of Mathematics, University of Maryland, Baltimore County

There has been increased interest recently in feed-back methods for reliable, robust, efficient computational methods in mechanics. We will outline the construction of such methods for two classes of problems, one describing special (anti-plane shear or ``plastic'' torsion) deformations of bars of rectangular or arched cross section, and one describing bending of plates for certain classes of loads. In particular, we will show how to reduce the dimension of the underlying problem ``adaptively''. For anti-plane shear of ``brittle'' or linear materials, this method is adaptive (optimal in rate of convergence). We shall emphasize the theoretical and computational aspects that have practical import to the performance of this method, such as the construction of a posteriori error estimators that are simple to compute, the selection of basis functions in the dimensional reduction -- ensuring optimal convergence rates in the ``thin'' limit as well as in the order of model increasing -- and the heuristic principle for extension. We will illustrate these concepts with computations. We then discuss the feasibility of efficient solution of the first couple of members of the sequence of models -- also with an eye towards reliability and adaptivity. Here we introduce and analyze stable discrete spaces with quasi-optimal approximation properties (with respect to increasing polynomial degree) as they pertain to some general classes of problems: scalar and systems of elliptic as well as semi-elliptic (Stokes') problems. The main technique is elliptic regularity over polyhedral domains and the modern treatment of saddle-point (not strictly coercive) problems. Time permitting, we will also discuss extensions of above to 3 dimensions and stability questions for first order operators in general.