A genuinely stable Lagrange–Galerkin scheme for convection-diffusion problems

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. For the P k -element we prove error estimates of O ( Δ t + h 2 + h k + 1 ) in ℓ ∞ ( L 2 ) -norm and of O ( Δ t + h 2 + h k ) in ℓ ∞ ( H 1 ) -norm. Numerical results reflect these estimates.