Preconditioning in the domain decomposition methods for the $p$-version with the hierarchical bases

A DD (domain decomposition) method is considered for the $p$- and $h$-$p$-versions with the curvilinear elements for the elliptic second order partial differential equation. Coordinate functions of the reference element are defined by the products of the Legendre's polynomials. Efficient DD preconditioner is suggested on the basis of preconditioners for the problems on subdomains and for the Shur complement.