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

We study the error estimates for the alternating evolution discontinuous Galerkin (AEDG) method to one-dimensional linear convectiondiffusion equations. The AEDG method for general convection-diffusion equations was introduced by H. Liu and M. Pollack [J. Comp. Phys. 307 (2016), 574-592], 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(h^k+1) for k-th degree polynomials, under the same stability condition with ∈ ~ h². For a fully discrete scheme with the forward Euler temporal discretization, we further obtain the L² error estimate of order O(τ +h^k+1), under the stability condition c₀τ ≤ ∈ < Qh² for time step τ and an error of order O(τ² + h^k+1) for the Crank-Nicolson time discretization with any time step τ. 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.