ESAIM: Mathematical Modelling and Numerical Analysis

Research Article

Robust a posteriori error estimates for finite element discretizations of the heat equation with discontinuous coefficients

Berrone, Stefano

Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy. sberrone@calvino.polito.it

Abstract

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.

(Received July 15 2005)

(Revised October 24 2006)

(Online publication February 15 2007)

Key Words:

  • A posteriori error estimates;
  • parabolic problems;
  • discontinuous coefficients.

Mathematics Subject Classification:

  • 65M60;
  • 65M15;
  • 65M50
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