Gradient recovery in adaptive finite element methods for parabolic problems

Abstract

We derive energy-norm aposteriori error bounds, using gradient recovery (ZZ) estimators to control the spatial error, for fully discrete schemes for the linear heat equation. This appears to be the first completely rigorous derivation of ZZ estimators for fully discrete schemes for evolution problems, without any restrictive assumption on the timestep size. An essential tool for the analysis is the elliptic reconstruction technique. Our theoretical results are backed with extensive numerical experimentation aimed at (a) testing the practical sharpness and asymptotic behaviour of the error estimator against the error, and (b) deriving an adaptive method based on our estimators. An extra novelty provided is an implementation of a coarsening error "preindicator", with a complete implementation guide in ALBERTA.

6 figures, 1 sketch, appendix with pseudocode

References in corpus (3)

• Elliptic reconstruction and a posteriori error estimates for fully discrete linear parabolic problems
• A posteriori error control for discontinuous Galerkin methods for parabolic problems
• A posteriori error estimates in the maximum norm for parabolic problems

Cited by in corpus (5)

• A posteriori error control for discontinuous Galerkin methods for parabolic problems
• A finite element method for fully nonlinear elliptic problems
• A comparison of duality and energy aposteriori estimates for L?(0,T;L2(Ω)) in parabolic problems
• Residual estimates for post-processors in elliptic problems
• Adaptive FEM with explicit time integration for the wave equation