  In this paper, we present and study $C^1$ Petrov-Galerkin and Gauss
collocation methods with arbitrary polynomial degree $k$ ($\ge 3$) for
one-dimensional elliptic equations. We prove that, the solution and its
derivative approximations converge with rate $2k-2$ at all grid points; and the
solution approximation is superconvergent at all interior roots of a special
Jacobi polynomial of degree $k+1$ in each element, the first-order derivative
approximation is superconvergent at all interior $k-2$ Lobatto points, and the
second-order derivative approximation is superconvergent at $k-1$ Gauss points,
with an order of $k+2$, $k+1$, and $k$, respectively. As a by-product, we prove
that both the Petrov-Galerkin solution and the Gauss collocation solution are
superconvergent towards a particular Jacobi projection of the exact solution in
$H^2$, $H^1$, and $L^2$ norms. All theoretical findings are confirmed by
numerical experiments.
