Abstract
We derive a posteriori error estimates for the discretization of the heat equation in a unified and fully discrete setting comprising the discontinuous Galerkin, various finite volume, and mixed finite element methods in space and the backward Euler scheme in time. Extensions to conforming and nonconforming finite element spatial discretizations are also outlined. Our estimates are based on a $H^1$-conforming reconstruction of the potential, continuous and piecewise affine in time, and a locally conservative $\mathbf{H}(\mathrm{div})$-conforming reconstruction of the flux, piecewise constant in time. They yield a guaranteed and fully computable upper bound on the error measured in the energy norm augmented by a dual norm of the time derivative. Local-in-time lower bounds are also derived; for nonconforming methods on time-varying meshes, the lower bounds require a mild parabolic-type constraint on the meshsize.
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