A C0 Interior Penalty Method for the Phase Field Crystal Equation

TL;DR

This paper introduces a novel C0 interior penalty finite element method for the sixth-order phase field crystal equation, demonstrating its stability, convergence, and effectiveness through theoretical analysis and numerical benchmarks.

Contribution

The paper develops the first C0 interior penalty finite element method for the phase field crystal equation, with detailed error analysis for time-dependent problems.

Findings

Method is uniquely solvable and unconditionally energy stable.

Error analysis methodology for time-dependent problems is established.

Numerical experiments confirm the method's effectiveness.

Abstract

We present a C0 interior penalty finite element method for the sixth-order phase field crystal equation. We demonstrate that the numerical scheme is uniquely solvable, unconditionally energy stable, and convergent. We remark that the novelty of this paper lies in the fact that this is the first C0 interior penalty finite element method developed for the phase field crystal equation. Additionally, the error analysis presented develops a detailed methodology for analyzing time dependent problems utilizing the C0 interior penalty method. We furthermore benchmark our method against numerical experiments previously established in the literature.