Analysis of generalised alternating local discontinuous Galerkin method on layer-adapted mesh for singularly perturbed problems

We study the generalised alternating local discontinuous Galerkin (GA-LDG) method for one- and two-dimensional singularly perturbed convection-diffusion problems. The method is equipped with an upwind-biased numerical flux for the convection term and a generalised alternating numerical flux for the diffusion term in the interior of the domain. For the one-dimensional case, we demonstrate an optimal uniform error estimate for the LDG method under the energy norm and an epsilon-weighted L-2 -norm. For the two-dimensional case, we establish an optimal or a quasi-optimal error estimate for the LDG method under the energy norm. Our results are valid for three typical layer-adapted meshes, namely the Shishkin mesh, the Bakhvalov-Shishkin mesh, and a Bakhvalov-type mesh. The findings of numerical experiments are presented to verify the theoretical results.