Abstract
Abstract We consider the a posteriori error analysis of fully discrete approximations of parabolic problems based on conforming $hp$-finite element methods in space and an arbitrary order discontinuous Galerkin method in time. Using an equilibrated flux reconstruction we present a posteriori error estimates yielding guaranteed upper bounds on the $L^2(H^1)$-norm of the error, without unknown constants and without restrictions on the spatial and temporal meshes. It is known from the literature that the analysis of the efficiency of the estimators represents a significant challenge for $L^2(H^1)$-norm estimates. Here we show that the estimator is bounded by the $L^2(H^1)$-norm of the error plus the temporal jumps under the one-sided parabolic condition $h^2 \lesssim \tau $. This result improves on earlier works that required stronger two-sided hypotheses such as $h \simeq \tau $ or $h^2\simeq \tau $; instead, our result now encompasses practically relevant cases for computations and allows for locally refined spatial meshes. The constants in our bounds are robust with respect to the mesh and time-step sizes, the spatial polynomial degrees and the refinement and coarsening between time steps, thereby removing any transition condition.
Highlights
We consider the heat equation∂tu − Δu = f in Ω × (0, T), u = 0 on ∂Ω × (0, T), (1.1)u(0) = u0 in Ω, A
Using an equilibrated flux reconstruction we present a posteriori error estimates yielding guaranteed upper bounds on the L2(H1)-norm of the error, without unknown constants and without restrictions on the spatial and temporal meshes
Our motivation for considering the heat equation (1.1) as a model problem is that the a posteriori error estimates developed in this context serve as a starting point for extensions to diverse applications, for example nonlinear problems
Summary
Ern et al (2017b) developed a posteriori error estimators, based on equilibrated fluxes, for arbitrary order discretizations of parabolic problems within the L2(H1) ∩ H1(H−1)-norm setting, that are guaranteed, locally efficient and robust. The estimators are robust with respect to the mesh and time-step sizes, and with respect to the polynomial degrees in space and time, and with respect to mesh coarsening and refinement, thereby removing the so-called transition conditions previously encountered in Verfürth (2003) These results are built upon the analysis for elliptic problems in Braess et al. Our main contributions, presented in Theorem 5.1 in Section 5 below, include guaranteed upper bounds for the L2(H1)-norm of the error and local-in-space-and-time lower bounds for the spatial estimator under the one-sided condition h2 τ. Throughout this paper, the notation a b means that a ≤ Cb, with a generic constant C that depends possibly on the shape-regularity of the spatial meshes and the space dimension d, but is otherwise independent of the mesh-size, time-step size, as well as the spatial and temporal polynomial degrees or on refinement and coarsening between time steps
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