  For a reaction-dominated diffusion problem we study a primal and a dual
hybrid finite element method where weak continuity conditions are enforced by
Lagrange multipliers. Uniform robustness of the discrete methods is achieved by
enriching the local discretization spaces with modified face bubble functions
which decay exponentially in the interior of an element depending on the ratio
of the singular perturbation parameter and the local mesh-size. A posteriori
error estimators are derived using Fortin operators. They are robust with
respect to the singular perturbation parameter. Numerical experiments are
presented that show that oscillations, if present, are significantly smaller
then those observed in common finite element methods.
