Discretizations of self-adjoint, linear, second-order, uniformly elliptic partial differential equations by hybrid mixed finite elements lead to large, ill-conditioned saddle-point problems. By eliminating the flux variable, a reduced problem is formed that is symmetric and positive definite but still large and ill-conditioned. Several domain decomposition and multigrid preconditioners are applied to the reduced problem, and bounds on their asymptotic rates of convergence are derived. Two Schwarz domain decomposition methods are shown to converge at least as fast asymptotically as the same methods applied to conforming linear finite element discretizations. In particular, for both the standard additive overlapping Schwarz method of Dryja and Widlund and one of the interfacial Schwarz methods of Smith, it is proven that the rates of convergence of the methods are uniformly bounded with respect to the mesh size in both two and three dimensions under standard assumptions. Several multigrid preconditioners are constructed for the reduced problem including a generalization of a method due to Bramble, Pasciak and Xu and an adaptation of methods of Wohlmuth and Hoppe. A common feature of these multigrid methods is the use of conforming finite element spaces on the coarser grids. Uniform convergence rates are proven for most of the methods and numerical results that verify the bounds are reported. A mixed finite element discretization of a simplified model of sediment transport in a two dimensional periodic channel is also described. The results of two simulations that employ one of the multigrid preconditioners are reported.