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 epsilon < Qh(2). Here epsilon is the method parameter, and h is the maximum spatial grid size. In this work, we establish optimal L-2 error estimates of order O(h(k+1)) for k-th degree polynomials, under the same stability condition with epsilon similar to h(2). For a fully discrete scheme with the forward Euler temporal discretization, we further obtain the L-2 error estimate of order O(tau+ h(k+1)), under the stability condition c0 tau <= epsilon < Qh(2) for time step tau; and an error of order O(tau(2) + h(k+1)) for the Crank-Nicolson time discretization with any time step t. 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.