Error Estimates for a Variable Time-Step Discretization of a Phase Transition Model with Hyperbolic Momentum

Abstract

This paper deals with a fully implicit time discretization scheme with variable time-step for a nonlinear system modelling phase transition and mechanical deformations in shape memory alloys. The model is studied in the nonstationary case and accounts for local microscopic interactions between the phases introducing the gradients of the phase parameters. The resulting initialboundary value problem has already been studied by the author who proved existence, uniqueness and continuous dependence on data for a suitable weak solution along some regularity results. A careful and detailed investigation of the variable time-step discretization is the goal of this paper. Thus, we deduce some estimates for the discretization error. These estimates depend only on data, impose no constraints between consecutive time-steps and show an optimal order of convergence. Finally, we prove another regularity result for the solution under stronger regularity assumptions on data.

Uniform attractors for a phase transition model coupling momentum balance and phase dynamics

“…Therefore, setting ψ(t) = e ε(t−τ )/2 ϕ(t) and integrating, it turns out that The argument of the proof relies on a time discretization -a priori estimates -passage to the limit procedure. Moreover, [27] contains the error estimates for the time discretization scheme approximating (2.19)-(2.21). The situation in which c > 0 can be analysed similarly, by simply adapting the proofs of [26] and [27].…”

“…Moreover, [27] contains the error estimates for the time discretization scheme approximating (2.19)-(2.21). The situation in which c > 0 can be analysed similarly, by simply adapting the proofs of [26] and [27]. The following statement holds.…”

Analysis of a time discretization scheme for a nonstandard viscous Cahn–Hilliard system

“…With reference to such a complete discretization of Cahn-Hilliard and viscous Cahn-Hilliard systems, we quote papers [1,2,3,4,5,6,7,8,22,21,23]. Some recent efforts can be found in the literature with the aim of analyzing other classes of phase transition problems, either to show existence via time discretization [9,14,15,19,20,27,30,35,36] or to prove numerical results such as special convergence properties, stability or error estimates [11,12,13,18,25,28,31,33,34] (cf. also [26] for a recent review on phase-field models).…”