In this work we derive a posteriori error estimates basedon equations residuals for the heat equation with discontinuousdiffusivity coefficients. The estimates are based on a fully discretescheme based on conforming finite elements in each time slab andon the A-stable θ-scheme with 1/2 ≤ θ ≤ 1.Following remarks of [Picasso, Comput. Methods Appl. Mech. Engrg. 167 (1998) 223–237; Verfürth, Calcolo40 (2003) 195–212] it is easyto identify a time-discretization error-estimatorand a space-discretization error-estimator. In this work we introduce a similarsplitting 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 boundswhose ratio is independent of any meshsize, timestep, problem parameter and its jumps.

We study the Dirichlet boundary value problem for eikonal type equations of ray light propagation in an inhomogeneous medium with discontinuous refraction index. We prove a comparison principle that allows us to obtain existence and uniqueness of a continuous viscosity solution when the Lie algebra generated by the coefficients satisfies a Hörmander type condition. We require the refraction index to be piecewise continuous across Lipschitz hypersurfaces. The results characterize the value function of the generalized minimum time problem with discontinuous running cost.