  In this paper, a new $C^1$-conforming Petrov-Galerkin method for
convection-diffusion equations is designed and analyzed.
  The trail space of the proposed method is a $C^1$-conforming ${\mathbb Q}_k$
(i.e., tensor product of polynomials of degree at most $k$) finite element
space while the test space is taken as the $L^2$ (discontinuous) piecewise
${\mathbb Q}_{k-2}$ polynomial space.
  Existence and uniqueness of the numerical solution is proved and optimal
error estimates in all $L^2, H^1, H^2$-norms are established.
  In addition, superconvergence properties of the new method are investigated
and superconvergence points/lines are identified at mesh nodes (with order
$2k-2$ for both function value and derivatives), at roots of a special Jacobi
polynomial, and at the Lobatto lines and Gauss lines with rigorous theoretical
analysis.
  In order to reduce the global regularity requirement, interior a priori error
estimates in the $L^2, H^1, H^2$-norms are derived. Numerical experiments are
presented to confirm theoretical findings.
