  We present a Lagrange-Galerkin scheme free from numerical quadrature for
convection-diffusion problems. Since the scheme can be implemented exactly as
it is, theoretical stability result is assured. While conventional
Lagrange-Galerkin schemes may encounter the instability caused by numerical
quadrature error, the present scheme is genuinely stable. We prove the
stability and convergence of the best possible order. Numerical results reflect
these estimates.
