  This paper provides the first provable $\mathcal{O}(N \log N)$ algorithms for
the linear system arising from the direct finite element discretization of the
fourth-order equation with different boundary conditions on unstructured grids
of size $N$ on an arbitrary polygoanl domain. Several preconditioners are
presented, and the conjugate gradient methods applied with these
preconditioners are proven to converge uniformly with respect to the size of
the preconditioned linear system. One main ingredient of the optimal
preconditioners is a mixed-form discretization of the fourth-order problem.
Such a mixed-form discretization leads to a non-desirable ---either non-optimal
or non-convergent--- approximation of the original solution, but it provides
optimal preconditioners for the direct finite element problem. It is further
shown that the implementation of the preconditioners can be reduced to the
solution of several discrete Poisson equations. Therefore, any existing optimal
or nearly optimal solver, such as geometric or algebraic multigrid methods, for
Poisson equations would lead to a nearly optimal solver for the discrete
fourth-order system. A number of nonstandard Sobolev spaces and their
discretizations defined on the boundary of polygonal domains are carefully
studied and used for the analysis of those preconditioners.
