  In this paper, we present a study on the effect of penalty term in the mixed
Discontinuous Galerkin Finite Element Method for the biharmonic equation
proposed by \cite{gudi2008mixed}. The proposed mixed Discontinuous Galerkin
Method showed sub-optimal rates of convergence for piecewise quadratic elements
and no significant convergence rates for piecewise linear elements. We show
that by choosing the penalty term proportional to $|e_k|^{-1}$ instead of
$|e_k|^{-3}$, ensures an optimal rate of convergence for the approximation,
including for piecewise linear elements. Finally, we present numerical
experiments to validate our theoretical results.
