Robusta posteriorierror estimates for finite element discretizations of the heat equation with discontinuous coefficients

Abstract

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 spacediscretization 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.

Skipping transition conditions ina posteriorierror estimates for finite element discretizations of parabolic equations

“…This introduces great difficulties in dealing with general adapted meshes. In [5] similar results were extended to the case of the heat equation with discontinuous, piecewise constant, coefficients.…”

“…In this section we derive a residual-based error estimator for the fully discretized model problem following the work in [5,17,20]. In particular, we shall derive global-in-space, local-in-time upper and lower bounds, as well as global-in-space, global-in-time upper and lower bounds.…”

“…First, we introduce some notation which will be used for the construction of the estimator, more detail can be found in [5]. …”

“…Following considerations of [17,20], and numerical experiments of [5], we can say that η n R is a space error estimator related to the quality of the partition T n h . When the mesh is suitably adapted, the quantity η n τ gives information on the error due to time discretization.…”

“…The a posteriori error estimates proposed can easily be extended to the case of piecewise constant discontinuous diffusivity coefficients assuming a quasi monotonicity condition [4,9,16] as in [5].…”