POINTWISE A POSTERIORI ERROR ESTIMATES FOR DISCONTINUOUS GALERKIN METHODS FOR SINGULARLY PERTURBED REACTION-DIFFUSION EQUATIONS^*

The symmetric interior penalty discontinuous Galerkin method and its version with weighted averages are considered on shape-regular nonconforming meshes with an arbitrarily large number of mesh faces contained in any element face. For this method, residual-type a posteriori error estimates in the maximum norm are given for singularly perturbed semilinear reaction-diffusion equations posed in polyhedral domains. The error constants are independent of the diameters of mesh elements and of the small perturbation parameter. The theoretical findings are illustrated by numerical experiments.