An unconditionally stable hybridizable-embedded discontinuous Galerkin method for the phase field crystal equation

TL;DR

This paper introduces a stable hybridizable-embedded discontinuous Galerkin method for the phase field crystal equation that simplifies computations and guarantees energy stability, demonstrated through numerical tests.

Contribution

It presents a novel unconditionally stable scheme that avoids high-order derivatives and reduces degrees of freedom for the phase field crystal equation.

Findings

Scheme is unconditionally energy stable.

Existence and uniqueness of solutions are proven.

Numerical examples confirm efficiency and accuracy.

Abstract

This paper presents a first-order convex splitting hybridizable/embedded discontinuous Galerkin method for the phase field crystal equation written in mixed form. Since the sixth-order phase field crystal equation is rewritten as a first-order system, our scheme avoids the calculation of high-order derivatives, which can be computationally expensive. The proposed method uses continuous facet unknowns and static condensation, which significantly reduces the number of coupled degrees of freedom. Using stabilization parameters that satisfy a simple and explicit relation, we show that our scheme is unconditionally energy stable. Moreover, we show the existence and uniqueness of the discrete solution for the case of variable mobility. The scheme's performance and properties are demonstrated through several numerical examples, including benchmark results that align with the existing literature, as well as a comparison of degrees of freedom against other methods.