  In this paper, we study the biharmonic equation with the Navier boundary
conditions in a polygonal domain. In particular, we propose a method that
effectively decouples the 4th-order problem into a system of Poisson equations.
Different from the usual mixed method that leads to two Poisson problems but
only applies to convex domains, the proposed decomposition involves a third
Poisson equation to confine the solution in the correct function space, and
therefore can be used in both convex and non-convex domains. A $C^0$ finite
element algorithm is in turn proposed to solve the resulted system. In
addition, we derive the optimal error estimates for the numerical solution on
both quasi-uniform meshes and graded meshes. Numerical test results are
presented to justify the theoretical findings.
