Maximum norm a posteriori error estimate for a 3d singularly perturbed semilinear reaction-diffusion problem

A singularly perturbed semilinear reaction-diffusion problem in the unit cube, is discretized on arbitrary nonuniform tensor-product meshes. We establish a second-order maximum norm a posteriori error estimate that holds true uniformly in the small diffusion parameter. No mesh aspect ratio condition is imposed. This result is obtained by combining (i) sharp bounds on the Green’s function of the continuous differential operator in the Sobolev W1,1 and W2,1 norms and (ii) a special representation of the residual in terms of an arbitrary current mesh and the current computed solution. Numerical results on a priori chosen meshes are presented that support our theoretical estimate.