A key component of DD (domain decomposition) solvers for hp discretizations of elliptic equations is the solver for the internal stiffness matrices of p-elements. We consider the algorithm of the linear complexity for solving such problems on spectral p-elements, which, therefore, in the leading DD solver plays the role of the second stage DD solver. It is based on the first order finite element preconditioning of the Orszag type for the reference element stiffness matrices. Earlier, for spectral elements, only fast solvers obtained with the use of special preconditioners in factored form were known. The most intricate part of the algorithm is the inter-subdomain Schur complement preconditioning by inexact iterative solver employing two preconditioners -- preconditioner-solver and preconditioner-multiplicator. From general point of view, the solver developed in the paper is the DD solver for the discretization on a strongly variable in size and shape deteriorating mesh with the number of subdomains growing with the growth of the number of degrees of freedom.