  In this paper, we study the sixth order equation with the simply supported
boundary conditions in a polygonal domain. We propose a new mixed formulation
that decomposes the sixth order problem into a system of Poisson equations.
Depending on the interior angles of the domain, additional Poisson problems may
be needed to confine the solution to the correct Sobolev space. In addition, we
propose a $C^0$ finite element algorithm for the sixth order problem and
provide the optimal error analysis. Numerical results are reported to verify
the theoretical findings.
