  The weak Galerkin (WG) finite element method is an effective and flexible
general numerical technique for solving partial differential equations. It is a
natural extension of the classic conforming finite element method for
discontinuous approximations, which maintains simple finite element
formulation. Stabilizer free weak Galerkin methods further simplify the WG
methods and reduce computational complexity. This paper explores the
possibility of optimal combination of polynomial spaces that minimize the
number of unknowns in the stabilizer free WG schemes without compromising the
accuracy of the numerical approximation. A new stabilizer free weak Galerkin
finite element method is proposed and analyzed with polynomial degree
reduction. To achieve such a goal, a new definition of weak gradient is
introduced. Error estimates of optimal order are established for the
corresponding WG approximations in both a discrete $H^1$ norm and the standard
$L^2$ norm. The numerical examples are tested on various meshes and confirm the
theory.
