Topological structures of boundary value problems in block elements

Block structures are considered; a boundary value problem for a system of inhomogeneous partial differential equations with constant coefficients is formulated in each block of a structure. The problem of matching solutions to boundary value problems in blocks with each other by topological study of the properties of solutions in the block structure is examined in the conditions of correct solvability of boundary value problems in blocks of the block structure. Some new properties of solutions to boundary value problems in block structures are found that are important for applications.     

Singular solutions of contact problems and block elements

In this work, we consider mixed problems of elasticity theory, in particular, contact problems for cases that are nontraditional. They include mixed problems with discontinuous boundary conditions in which the singularities in the behavior of contact stresses are not studied or the energy of the singularities is unbounded. An example of such mixed problems is contact problems for two rigid stamps approaching each other by rectilinear boundaries up to contact but not merging into one stamp. It has been shown that such problems, which appear in seismology, failure theory, and civil engineering, have singular components with unbounded energy and can be solved by topological methods with pointwise convergence, in particular, by the block element method. Numerical methods that are based on using the energy integral are not applicable to such problems in view of its divergence.