A class of linear parabolic equations is considered. We derive a common framework for the a posteriori error analysis of certain second-order time discretizations combined with finite element discretizations in space. In particular, we study the Crank–Nicolson method, the extrapolated Euler method, the backward differentiation formula of order 2, the Lobatto IIIC method and a two-stage SDIRK method. We use the idea of elliptic reconstructions and certain bounds for the Green’s function of the parabolic operator.
In the present paper we analyse a finite element method for a singularly perturbed convection-diffusion problem with exponential boundary layers. Using a mortaring technique we combine an anisotropic triangulation of the layer region (into rectangles) with a shape regular one of the remainder of the domain. This results in a possibly non-matching (and hybrid), but layer adapted mesh of Shishkin type. We study the error of the method allowing different asymptotic behaviour of the triangulations and prove uniform convergence and a supercloseness property of the method. Numerical results supporting our analysis are presented.
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