Singular and hierarchical vector functions for multiscale problems

A number of new hierarchical and singular (scalar and vector) basis functions for field representation in two- and three-dimensional cells have been recently developed and discussed in a set of different papers, and in a new book. The hierarchical functions are designed for adaptive p-refinement on domain meshed with differently shaped cells. The functions guarantee built-in continuity on the common boundaries of adjacent cells for the tangent and normal vector component in the curl- and in the divergence-conforming case, respectively. The polynomial order of the functions may vary from cell to cell to facilitate application to multiscale problems, and to permit one to use cells of large size. At the oral presentation we will review these functions and their key features, and present new results obtained with their use.