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.