  In this paper, we present a high-order finite element method based on a
reconstructed approximation to the biharmonic equation. In our construction,
the space is reconstructed from nodal values by solving a local least squares
fitting problem per element. It is shown that the space can achieve an
arbitrarily high-order accuracy and share the same nodal degrees of freedom
with the $C^0$ linear space. The interior penalty discontinuous Galerkin scheme
can be directly applied to the reconstructed space for solving the biharmonic
equation. We prove that the numerical solution converges with optimal orders
under error measurements. More importantly, we establish a norm equivalence
between the reconstructed space and the continuous linear space. This property
allows us to precondition the linear system arising from the high-order space
by the linear space on the same mesh. This preconditioner is shown to be
optimal in the sense that the condition number of the preconditioned system
admits a uniform upper bound independent of the mesh size. Numerical examples
in two and three dimensions are provided to illustrate the accuracy of the
scheme and the efficiency of the preconditioning method.
