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

摘要

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 theta-scheme with 1/ 2 <=theta <= 1. Following remarks of [ Picasso, Comput. Methods Appl. Mech. Engrg. 167 ( 1998) 223 - 237; Verfurth, 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.