The hp- and h-versions of the discontinuous and local discontinuous Galerkin methods for one-dimensional singularly perturbed models

We study the numerical solution of a class of singularly perturbed models in one dimension by discontinuous Galerkin (DG) and local DG (LDG) methods. Using an hp-version DG method, we show that exponential rates of convergence can be achieved for solutions of singularly perturbed first order problems with inflow boundary layers caused by the diffusion parameter ϵ. Moreover, we prove that by employing a graded mesh of Shishkin type, algebraic O ( ( log N / N ) p + 1 ) convergence rates can be achieved for the h-version DG method when polynomials of degree at most p are used, where N is the number of mesh subintervals. Similar results have been shown by applying hp- and h-versions of the LDG method for a class of one-dimensional convection–diffusion problems with outflow boundary layers.