EFFECTIVE NUMERICAL METHODS FOR SOLVING 
               ELLIPTIC PROBLEMS IN STRENGTHENED SOBOLEV SPACES

                             Eugene G. D'yakonov 
             Department of Computer Mathematics and Cybernetics 
                           Moscow State University 
                           Moscow, 119899, Russia 

                                   Abstract

Fourth-order elliptic boundary value problems in the plane can be reduced to
operator equations in Hilbert spaces $G$ that are certain subspaces of the
Sobolev space $W_2^2(\Omega)\equiv G^{(2)}$.  Appearance of asymptotically
optimal algorithms for Stokes type problems made it natural to focus on an
approach that considers rot $w\equiv[D_2w,-D_1w]\equiv \vec u$ as a new
unknown vector-function, which automatically satisfies the condition
div $\vec u=0$.  In this work, we show that this approach can also be
developed for an important class of problems from the theory of plates and
shells with stiffeners.  The main mathematical problem was to show that the
well-known inf-sup condition (normal solvability of the divergence operator)
holds for special Hilbert spaces.  This result is also essential for certain
hydrodynamics problems.