Operator problems in strengthened Sobolev spaces and numerical methods for them

Abstract

The strengthened Sobolev spaces are naturally connected, e.g., with such important (two or three-dimensional) problems of mathematical physics as those in theory of plates and shells with stiffeners or in the capillary hydrodynamics involving the surface tension. These nonstandard Hilbert spaces allow also to set variational and operator problems on composed manifolds of different dimensionality. Spectral (eigenvalue) problems can be considered as well.

Special attention is paid to numerical methods based on the use of projective-grid methods and effective iterative methods such as multigrid and cutting methods; under natural conditions on the smoothness of the solution, it can be shown that the strengthened variant of the Kolmogorov-Bakhvalov hypothesis about asymptotically optimal algorithms for elliptic problems holds also for the above mentioned problems on composed manifolds.