Equilibrated solutions, locally satisfying all the equilibrium conditions, may be obtained by using a special case of the hybrid finite element formulation. Equilibrium finite element solutions will normally present compatibility defaults, which may be directly used to estimate the error of the solution, a posteriori. Another approach is to construct a compatible solution using the stresses and displacements available from the hybrid solution. From this dual solution, an upper bound for the global error is obtained. In this paper, the hybrid equilibrium element formulation, the occurrence of spurious kinematic modes and the use of super-elements, in 2D and 3D, are briefly reviewed. Compatibility defaults for 2D and 3D are presented, together with an expression for an element error indicator explicitly based on such defaults. A local procedure for recovering conforming displacements from the equilibrium finite element solution is also presented. The h-refinement procedure is adapted to prevent irregular refinement patterns.