  In this paper, we derive improved a priori error estimates for families of
hybridizable interior penalty discontinuous Galerkin (H-IP) methods using a
variable penalty for second-order elliptic problems. The strategy is to use a
penalization function of the form $\mathcal{O}(1/h^{1+\delta})$, where $h$
denotes the mesh size and $\delta$ is a user-dependent parameter. We then
quantify its direct impact on the convergence analysis, namely, the (strong)
consistency, discrete coercivity, and boundedness (with
$h^{\delta}$-dependency), and we derive updated error estimates for both
discrete energy- and $L^{2}$-norms. The originality of the error analysis
relies specifically on the use of conforming interpolants of the exact
solution. All theoretical results are supported by numerical evidence.
