Error estimates for the AEDG method to one-dimensional linear convection-diffusion equations

Abstract. We study the error estimates for the alternating evolution discontinuous Galerkin (AEDG) method to one dimensional linear convection-diÆusion equations. The AEDG method for general convection-diÆusion equations was introduced in [H. Liu, M. Pollack, J. Comp. Phys. 307: 574–592, 2016], where stability of the semi-discrete scheme was rigorously proved for linear problems under a CFL-like stability condition ≤ < Qh2. Here ≤ is the method parameter, and h is the maximum spatial grid size. In this work, we establish optimal L2 error estimates of order O(hk+1) for k-th degree polynomials, under the same stability condition with ≤ a h2. For fully discrete scheme with the forward Euler temporal discretization, we further obtain the L2 error estimate of order O(ø + hk+1), under the stability condition c0ø ∑ ≤ < Qh2 for time step ø ; and error of order O(ø2 + hk+1) for the Crank-Nicolson time discretization with any time step ø . Key tools include two approximation spaces to distinguish overlapping polynomials, two bi-linear operators, coupled global projections, and a duality argument adapted to the situation with overlapping polynomials.