We introduce a novel minimal order hybrid Discontinuous Galerkin (HDG) and a
novel mass conserving mixed stress (MCS) method for the approximation of
incompressible flows. For this we employ the H(div)-conforming
linear Brezzi-Douglas-Marini space and the lowest order Raviart-Thomas space
for the approximation of the velocity and the vorticity, respectively. Our
methods are based on the physically correct diffusive flux −νε(u) and provide exactly divergence-free discrete velocity
solutions, optimal (pressure robust) error estimates and a minimal number of
coupling degrees of freedom. For the stability analysis we introduce a new
Korn-like inequality for vector-valued element-wise H1 and normal continuous
functions. Numerical examples conclude the work where the theoretical findings
are validated and the novel methods are compared in terms of condition numbers
with respect to discrete stability parameters