A comparison of duality and energy aposteriori estimates for L?(0,T;L2(Ω)) in parabolic problems

Abstract

We use the elliptic reconstruction technique in combination with a duality approach to prove aposteriori error estimates for fully discrete back- ward Euler scheme for linear parabolic equations. As an application, we com- bine our result with the residual based estimators from the aposteriori esti- mation for elliptic problems to derive space-error indicators and thus a fully practical version of the estimators bounding the error in the L \infty (0, T ; L2(Ω)) norm. These estimators, which are of optimal order, extend those introduced by Eriksson and Johnson (1991) by taking into account the error induced by the mesh changes and allowing for a more flexible use of the elliptic estima- tors. For comparison with previous results we derive also an energy-based aposteriori estimate for the L \infty (0, T ; L2(Ω))-error which simplifies a previous one given in Lakkis and Makridakis (2006). We then compare both estimators (duality vs. energy) in practical situations and draw conclusions.

30 pages, including 7 color plates in 4 figures

References in corpus (4)

• 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
• Gradient recovery in adaptive finite element methods for parabolic problems

Cited by in corpus (4)

• Guaranteed, locally space-time efficient, and polynomial-degree robust a posteriori error estimates for high-order discretizations of parabolic problems
• Stability of step size control based on a posteriori error estimates
• An introduction to the a posteriori error analysis of parabolic partial differential equations
• On the efficiency of a posteriori error estimators for parabolic partial differential equations in the energy norm