In this work some particular fast prototype iterative preconditioned solvers are designed for a semi-hybridization approach of stable high-order mixed finite element formulations of Darcy’s flows. The main issue consists in exploiting properties of an available hierarchy of H(div)-conforming shape functions, specially of recently constructed divergence-free ones. For that, in the flux space of trace type, two parts are considered in terms of the corresponding normal components over and element boundary: one of piecewise constants and the second for higher order functions with zero integral. The lower order flux is hybridized by a constant pressure per interface element (Lagrange multiplier). This way, the system of equations is condensed to a global problem by means of a 2 × 2 block system associated with the two kinds of flux functions of trace type. In terms of conditioning, the strategy has similar characteristics as the algebraic Primal Iterative Substructuring schemes. The application of Conjugate Gradient iterations and a low order preconditioner to solve higher-order problems reveals interesting characteristics: a conservative solution is obtained at each step of the iterative scheme and the number of iterations needed to convergence is independent of the number of elements. Thus, the process can be interrupted at any step, the partial solution being still valid for applications, e. g. in transport problems. Polylogarithmic bound for the matrix condition number κ=O(1+log(k))2 has also been accessed for both 2D and 3D numerical tests.
