Discontinuous Galerkin methods with interior penalties on graded meshes for 2D singularly perturbed convection–diffusion problems

Abstract

In this paper, we introduce discontinuous Galerkin methods with interior penalties, both the NIPG and SIPG method for solving 2D singularly perturbed convection–diffusion problems. On the modified graded meshes with the standard Lagrange Qk-elements (k=1,2), we show optimal order error estimates in the ε-weighted energy norm uniformly, up to a logarithmic factor, in the singular perturbation parameter ε. We prove that the convergence rate in the ε-weighted energy norm is O(logk+1⁡(1ε)Nk), where the total number of the mesh points is O(N2). For k≥3, our methods can be extended directly, provided the higher order regularities of the solution u are derived. Finally, numerical experiments support our theoretical results.