Mathematical Sciences Department
Analysis/PDE Seminar
Speaker: Olena Burkovska, Oak Ridge National Labs
Monday, November 20, 2023
11:00 am - 12:00 pm
Virtual, via Zoom: 929 6655 1992
Host: Nicole Buczkowski
Title: Non-isothermal nonlocal phase-field models permitting sharp interfaces
Abstract: In this talk, we will present non-isothermal nonlocal phase-field models of Cahn-Hilliard and AllenCahn types involving a nonsmooth double-obstacle potential. We demonstrate that under certain conditions and with a careful choice of the nonlocal operator, one can obtain a model that allows for sharp interfaces in the solution with only pure phases that evolve in time. This is a desirable property in many applications, such as the solidification of pure materials, and can be contrasted to the diffuse-interface local models that can not resolve sharp interfaces. Mathematically, in a weak form, the model translates to a system of variational inequalities coupled to a temperature evolution equation. We present the well-posedness analysis of the models and discuss appropriate discretization methods that can be realized efficiently. More specifically, we provide first and second-order timestepping schemes with the finite element or spectral approximations in spatial dimension. We show that a particular structure of the model and nonlocal kernel allows for the characterization of the solution in terms of a projection formula. For some cases, this, together with an appropriate timestepping scheme, provides a fast and efficient way to evaluate the solution without having to solve both a nonlinear and nonlocal system.