An introduction to the a posteriori error analysis of parabolic partial differential equations

Abstract

This article provides a brief introduction to the a posteriori error analysis of parabolic partial differential equations, with an emphasis on challenges distinct from those of steady-state problems. Using the heat equation as a model problem, we examine the crucial influence of the choice of error norm, as well as the choice of notion of reconstruction of the discrete solution, on the analytical properties of the resulting estimators, especially in terms of the efficiency of the estimators.

References in corpus (11)

• Elliptic reconstruction and a posteriori error estimates for fully discrete linear parabolic problems
• On Bogovski\uı and regularized Poincaré integral operators for de Rham complexes on Lipschitz domains
• A posteriori error control for discontinuous Galerkin methods for parabolic problems
• A posteriori error estimates in the maximum norm for parabolic problems
• Guaranteed, locally space-time efficient, and polynomial-degree robust a posteriori error estimates for high-order discretizations of parabolic problems
• Stable broken $H^1$ and $\mathbf{H}(\mathrm{div})$ polynomial extensions for polynomial-degree-robust potential and flux reconstruction in three space dimensions
• Discrete $p$-robust $\mathbf{H}(\mathrm{div})$-liftings and a posteriori estimates for elliptic problems with $H^{-1}$ source terms
• Simple and robust equilibrated flux a posteriori estimates for singularly perturbed reaction-diffusion problems
• Equilibrated flux a posteriori error estimates in $L^2(H^1)$-norms for high-order discretizations of parabolic problems
• A comparison of duality and energy aposteriori estimates for L?(0,T;L2(Ω)) in parabolic problems
• Finite element approximation of time-dependent mean field games with nondifferentiable Hamiltonians