Analysis of a Multiscale Discontinuous Galerkin Method for Convection‐Diffusion Problems

We study a multiscale discontinuous Galerkin method introduced in [T. J. R. Hughes, G. Scovazzi, P. Bochev, and A. Buffa, Comput. Meth. Appl. Mech. Engrg., 195 (2006), pp. 2761–2787] that reduces the computational complexity of the discontinuous Galerkin method, seemingly without adversely affecting the quality of results. For a stabilized variant we are able to obtain the same error estimates for the convection‐diffusion equation as for the usual discontinuous Galerkin method. We assess the stability of the unstabilized case numerically and find that the inf‐sup constant is positive, bounded uniformly away from zero, and very similar to that for the usual discontinuous Galerkin method.