Maximum-norm a posteriori error estimates for singularly perturbed elliptic reaction-diffusion problems

Residual-type a posteriori error estimates in the maximum norm are given for singularly perturbed semilinear reaction-diffusion equations posed in polyhedral domains. Standard finite element approximations are considered. The error constants are independent of the diameters of mesh elements and the small perturbation parameter. In our analysis, we employ sharp bounds on the Green's function of the linearized differential operator. Numerical results are presented that support our theoretical findings.