A Minimization Method for The Double-Well Energy Functional

TL;DR

This paper introduces an iterative minimization method for the double-well energy functional in phase-field theory, demonstrating unconditional energy stability and deriving a stable scheme for the Allen-Cahn equation.

Contribution

It proposes a new minimization approach for the double-well energy functional and develops a stable scheme for the Allen-Cahn equation based on Invariant Energy Quadratization.

Findings

Method is unconditionally energy stable

Derived a variant of the first-order scheme for Allen-Cahn

Proved unconditional energy stability of the proposed schemes

Abstract

In this paper an iterative minimization method is proposed to approximate the minimizer to the double-well energy functional arising in the phase-field theory. The method is based on a quadratic functional posed over a nonempty closed convex set and is shown to be unconditionally energy stable. By the minimization approach, we also derive an variant of the first-order scheme for the Allen-Cahn equation, which has been constructed in the context of Invariant Energy Quadratization, and prove its unconditional energy stability.