A POSTERIORI ERROR ESTIMATION FOR PARABOLIC PROBLEMS USING ELLIPTIC RECONSTRUCTIONS. I: BACKWARD-EULER AND CRANK-NICOLSON METHODS∗

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. Fully discrete Bakward-Euler and Crank-Nicolson methods are addressed, for which we employ elliptic reconstructions that are, respectively, piecewise-constant and piecewiselinear in time. We also use certain bounds for the Green’s function of the parabolic operator.

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