On the local discontinuous Galerkin method for singularly perturbed problem with two parameters

The local discontinuous Galerkin method is studied for a singularly perturbed problem with two parameters. We use a layer-adapted mesh in terms of mesh generating functions, which are discussed in an abstract form and satisfy several assumptions. Robust or almost robust convergence is established in different norms including the energy, L2, balanced, and maximum norms. Our analysis is based on several projections, i.e., different combinations of local L2 and local Gauss–Radau projections. By careful approximation error estimation on anisotropic meshes, we establish different error estimates in the mentioned norms. Numerical experiments are also included to test the theoretical results.