Authors’ abstract: We introduce a new mixed method for the biharmonic problem. The method is based on a formulation, where the biharmonic problem is rewritten as a system of four first-order equations. A hybrid form of the method is introduced, which allows us to reduce the globally coupled degrees of freedom to only those associated with Lagrange multipliers, which approximate the solution and its derivative at the faces of the triangulation. For

$k\ge 1$, a projection of the primal variable error superconverges with order

$k+3$, while the error itself converges with order

$k+1$ only. This fact is exploited by using local postprocessing techniques that produce new approximations to the primal variable converging with order

$k+3$. We provide numerical experiments that validate our theoretical results.