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 non-stationary case and accounts for local microscopic interactions between the phases introducing the gradients of the phase parameters. The resulting initial-boundary 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.

Key Words:

• Phase transitions
• Time discretization
• Convergence results
• Error control

Mathematics Subject Classification:

• 74N15
• 35Q72
• 65M15
• 74H15
• 80A22

Acknowledgment

This work has been partially supported by Istituto di Matematica Applicata e Tecnologie Informatiche-CNR, Pavia, Italy