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.