Robust a posteriori error estimates for finite element discretizations of the heat equation with discontinuous coefficients
Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy. firstname.lastname@example.org
Revised: 24 October 2006
In this work we derive a posteriori error estimates based on equations residuals for the heat equation with discontinuous diffusivity coefficients. The estimates are based on a fully discrete scheme based on conforming finite elements in each time slab and on the A-stable θ-scheme with 1/2 ≤ θ ≤ 1. Following remarks of [Picasso, Comput. Methods Appl. Mech. Engrg. 167 (1998) 223–237; Verfürth, Calcolo 40 (2003) 195–212] it is easy to identify a time-discretization error-estimator and a space-discretization error-estimator. In this work we introduce a similar splitting for the data-approximation error in time and in space. Assuming the quasi-monotonicity condition [Dryja et al., Numer. Math. 72 (1996) 313–348; Petzoldt, Adv. Comput. Math. 16 (2002) 47–75] we have upper and lower bounds whose ratio is independent of any meshsize, timestep, problem parameter and its jumps.
Mathematics Subject Classification: 65M60 / 65M15 / 65M50
Key words: A posteriori error estimates / parabolic problems / discontinuous coefficients.
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