Maximum norm a posteriori error estimation for parabolic problems using elliptic reconstructions

A semilinear second-order parabolic equation is considered in a regular and a singularly perturbed regime. For this equation, we give computable a posteriori error estimates in the maximum norm. Semidiscrete and fully discrete versions of the backward Euler, Crank--Nicolson, and discontinuous Galerkin $dG(r)$ methods are addressed. For their full discretizations, we employ elliptic reconstructions that are, respectively, piecewise-constant, piecewise-linear, and piecewise-quadratic for $r=1$ in time. We also use certain bounds for the Green's function of the parabolic operator.