Approximation of a nonlinear thermoelastic problem with a moving boundary via a fixed-domain method

Abstract

The thermoelastic stresses created in a solid phase domain in the course of solidification of a molten ingot are investigated. A nonlinear behaviour of the solid phase is admitted, too. This problem, obtained from a real situation by many simplifications, contains a moving boundary between the solid and the liquid phase domains. To make the usage of standard numerical packages possible, we propose here a fixed-domain approximation by means of including the liquid phase domain into the problem (in this way we get the fixed domain involving the whole ingot) and by replacing the liquid phase with a solid phase having, however, a small shear modulus. The weak $L^2$-convergence of thus approximated stresses in the solid phase domain is demonstrated. Besides, this convergence is shown to be strong on subsets whose closure belongs to the solid phase domain.