  This article investigates a local discontinuous Galerkin (LDG) method for
one-dimensional and two-dimensional singularly perturbed reaction-diffusion
problems on a Shishkin mesh. During this process, due to the inability of the
energy norm to fully capture the behavior of the boundary layers appearing in
the solutions, a balanced norm is introduced. By designing novel numerical
fluxes and constructing special interpolations, optimal convergences under the
balanced norm are achieved in both 1D and 2D cases. Numerical experiments
support the main theoretical conclusions.
