Error analysis of discontinuous Galerkin methods on layer adapted meshes for two dimensional turning point problem

A class of two-dimensional singularly perturbed convection–diffusion problems with turning points is studied in this work. Since the turning point takes place inside the domain, the problem usually exhibits a weak interior layer of cusp-type. The inclusion of parabolic boundary layers in the solution is caused by the presence of a small perturbation parameter. Two methods are designed to outperform the error estimation: the non-symmetric discontinuous Galerkin technique with an interior penalty (NIPG) and the symmetric discontinuous Galerkin method with an interior penalty (SIPG). To circumvent the layering effect in domain discretization, the traditional Shishkin mesh is used. Uniform error estimates are derived in the respective norms of order \(\mathcal {O}(N^{-1} \ln ^{3/2} N)\) $\mathcal{O}(N^{-1}{ln}^{3/2}N)$ and \(\mathcal {O}(N^{-1} \ln ^{2} N)\) $\mathcal{O}(N^{-1}{ln}^{2}N)$ for the NIPG and the SIPG methods, respectively. Numerical experiments are carried out to validate theoretical findings.