Uniform convergence of a coupled method for convection-diffusion problems in 2D Shishkin mesh (Q2865647)

The authors consider a convection-diffusion problem (with constant diffusion \(\epsilon\)) in a square along with homogeneous Dirichlet boundary conditions and propose, for its numerical solution, on a Shishkin mesh a coupling of linear finite elements (in the boundary layer region) and a nonsymmetric inner penalty discontinuous Galerkin method (in the rest of the square). Following \textit{D. N. Arnold} et al. [SIAM J. Numer. Anal. 39, No. 5, 1749--1779 (2002; Zbl 1008.65080)], and defining appropriate (upstream) numerical fluxes, they show the stability of the obtained coupled approach. Using compatibility conditions and results of \textit{T. Linß\ } and \textit{M. Stynes} [J. Math. Anal. Appl. 261, No. 2, 604--632 (2001; Zbl 1200.35046)], and others to circumvent the use of a decomposition of the analytical solution, they show \(\epsilon\)-uniform convergence of order \(N^{-1}\ln N\) in the \(\epsilon\)-norm. Illustrating their result by numerical experiments, they conclude that an \(L_2\)-estimate of order \(N^{-2}\) should also hold.