  In this article, we present and analyze a stabilizer-free $C^0$ weak Galerkin
(SF-C0WG) method for solving the biharmonic problem. The SF-C0WG method is
formulated in terms of cell unknowns which are $C^0$ continuous piecewise
polynomials of degree $k+2$ with $k\geq 0$ and in terms of face unknowns which
are discontinuous piecewise polynomials of degree $k+1$. The formulation of
this SF-C0WG method is without the stabilized or penalty term and is as simple
as the $C^1$ conforming finite element scheme of the biharmonic problem.
Optimal order error estimates in a discrete $H^2$-like norm and the $H^1$ norm
for $k\geq 0$ are established for the corresponding WG finite element
solutions. Error estimates in the $L^2$ norm are also derived with an optimal
order of convergence for $k>0$ and sub-optimal order of convergence for $k=0$.
Numerical experiments are shown to confirm the theoretical results.

Numerical Analysis;Numerical Analysis