  A new weak Galerkin (WG) finite element method for solving the biharmonic
equation in two or three dimensional spaces by using polynomials of reduced
order is introduced and analyzed. The WG method is on the use of weak functions
and their weak derivatives defined as distributions. Weak functions and weak
derivatives can be approximated by polynomials with various degrees. Different
combination of polynomial spaces leads to different WG finite element methods,
which makes WG methods highly flexible and efficient in practical computation.
This paper explores the possibility of optimal combination of polynomial spaces
that minimize the number of unknowns in the numerical scheme, yet without
compromising the accuracy of the numerical approximation. Error estimates of
optimal order are established for the corresponding WG approximations in both a
discrete $H^2$ norm and the standard $L^2$ norm. In addition, the paper also
presents some numerical experiments to demonstrate the power of the WG method.
The numerical results show a great promise of the robustness, reliability,
flexibility and accuracy of the WG method.
