  We present the lowest-order hybridizable discontinuous Galerkin schemes with
numerical integration (quadrature), denoted as HDG-P0, for the
reaction-diffusion equation and the generalized Stokes equations on conforming
simplicial meshes in two- and three-dimensions. Here by lowest order, we mean
that the (hybrid) finite element space for the global HDG facet degrees of
freedom (DOFs) is the space of piecewise constants on the mesh skeleton. A
discontinuous piecewise linear space is used for the approximation of the local
primal unknowns. We give the optimal a priori error analysis of the proposed
{\sf HDG-P0} schemes, which hasn't appeared in the literature yet for HDG
discretizations as far as numerical integration is concerned.
  Moreover, we propose optimal geometric multigrid preconditioners for the
statically condensed HDG-P0 linear systems on conforming simplicial meshes. In
both cases, we first establish the equivalence of the statically condensed HDG
system with a (slightly modified) nonconforming Crouzeix-Raviart (CR)
discretization, where the global (piecewise-constant) HDG finite element space
on the mesh skeleton has a natural one-to-one correspondence to the
nonconforming CR (piecewise-linear) finite element space that live on the whole
mesh. This equivalence then allows us to use the well-established nonconforming
geometry multigrid theory to precondition the condensed HDG system. Numerical
results in two- and three-dimensions are presented to verify our theoretical
findings.
