On diffusion-uniform error estimates for the DG method applied to singularly perturbed problems

This paper is concerned with the analysis of the discontinuous Galerkin (DG) finite element method applied to a nonstationary nonlinear convection–diffusion problem on quasi-uniform triangulations. Using the technique of Zhang & Shu (2004, SIAM J. Numer. Anal., 42, 641–666), we prove a priori error estimates which are uniform with respect to the diffusion coefficient ϵ→0 and valid even in the purely convective case. Zhang and Shu perform their analysis for various explicit schemes using an argument which relies heavily on mathematical induction. We extend the analysis to the method of lines using continuous mathematical induction and a nonlinear Gronwall-type lemma. For an implicit scheme, we prove that standard arguments cannot prove the desired estimates without additional assumptions. For this purpose, we use a suitable continuation of the discrete implicit solution and again use continuous mathematical induction to prove error estimates under a CFL-like condition. Finally, we extend the analysis from globally Lipschitz continuous convective nonlinearities to the locally Lipschitz continuous case.