Duality-based two-level error estimation for time-dependent PDEs: application to linear and nonlinear parabolic equations
Şimşek, G. and Wu, X. and van der Zee, K.G. and van Brummelen, E.H. (2015) Duality-based two-level error estimation for time-dependent PDEs: application to linear and nonlinear parabolic equations. Computer Methods in Applied Mechanics and Engineering, 288 . pp. 83-109. ISSN 1879-2138
We introduce a duality-based two-level error estimator for linear and nonlinear time-dependent problems. The error measure can be a space-time norm, energy norm, final-time error or other error related functional. The general methodology is developed for an abstract nonlinear parabolic PDE and subsequently applied to linear heat and nonlinear Cahn-Hilliard equations. The error due to finite element approximations is estimated with a residual weighted approximate-dual solution which is computed with two primal approximations at nested levels. We prove that the exact error is estimated by our estimator up to higher-order remainder terms. Numerical experiments confirm the theory regarding consistency of the dual-based two-level estimator. We also present a novel space-time adaptive strategy to control errors based on the new estimator.
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