An inverse problem for the one-phase Stefan problem with varying melting temperature

TL;DR

This paper addresses both the forward and backward solutions of a one-phase Stefan problem with a time-varying melting temperature, developing numerical algorithms to simulate and reconstruct the melting process based on evolving geometry.

Contribution

It introduces novel numerical algorithms for solving the inverse Stefan problem with variable melting temperature, including a moving mesh finite element method.

Findings

Successfully computed the evolution of the domain with varying melting temperature.

Reconstructed the time-dependent melting temperature from geometric data.

Validated algorithms through numerical simulations.

Abstract

The present article is dedicated to the forward and backward solution of a transient one-phase Stefan problem. In the forward problem, we compute the evolution of the initial domain for a Stefan problem where the melting temperature varies over time. This occurs in practice, for example, when the pressure in the external space changes in time. In the corresponding backward problem, we then reconstruct the time-dependent melting temperature from the knowledge of the evolving geometry. We develop respective numerical algorithms using a moving mesh finite element method and provide numerical simulations.