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

Kopteva, Natalia and Linss, Torsten (2013) Maximum norm a posteriori error estimation for parabolic problems using elliptic reconstructions. SIAM Journal on Numerical Analysis, 51 (3). pp. 1494-1524. ISSN 0036-1429

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    Abstract

    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.