Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Giselle Saylor, Tamas L. Horvath, Natasha S. Sharma

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.

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

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.
Paper Structure (14 sections, 8 theorems, 96 equations, 7 figures, 3 tables)

This paper contains 14 sections, 8 theorems, 96 equations, 7 figures, 3 tables.

Table of Contents

  1. Introduction
  2. The hybridizable/embedded discontinuous Galerkin discretization
  3. Energy stability
  4. First-order convex splitting temporal discretization
  5. Existence of the solution
  6. Numerical results
  7. Accuracy of the scheme
  8. Growth of a Monocrystal in a supercooled liquid
  9. Benchmark
  10. Polycrystal Growth and Grain Boundary Dynamics
  11. Phase transition
  12. Comparison of degrees of freedom
  13. Energy stability
  14. Conclusions and future work

Key Result

Lemma 3.1

If $\tau_1 > 0$, $\tau_2 = \tau_1$, $\tau_3 > 0$ and $\tau_4 = 2\tau_1$ in the definition of the numerical fluxes eq:num_fluxes, then the semi-discrete scheme satisfies the following energy balance where the semi-discrete energy $E_h$ is given by

Figures (7)

  • Figure 1: The error as a function of the number of degrees of freedom. The error values are taken from \ref{['tab:periodic']}, for $\Delta t / h = 0.95$. Continuous line EDG, dashed line HDG discretization.
  • Figure 2: Crystal growth on a rectangular domain $\left(0, \frac{36\pi}{\sqrt{3}}\right) \times \left(0, 24\pi\right)$ using a mesh consisting of $460 \times 532$ elements, and a time step size $\Delta t = 0.01$. The times shown are $t=0, 20, 40, 60, 80$ (from left to right, top to bottom). The colors are scaled from $\phi = -0.67$ to $\phi = 0.72$. See \ref{['ss:crystal_formation']} for more details.
  • Figure 3: Benchmark test case, the decrease of the total scaled energy.
  • Figure 4: Benchmark test case, output at different times. The times shown are $t=0, 5, 10$ (from left to right). The domain is $(0,32)^2$, the mesh consists of $256 \times 256$ elements, and $\Delta t = 0.005$. The colors are scaled from a density value of $\phi = 0.04$ to $\phi = 0.096$. See \ref{['ss:benchmark']} for more details.
  • Figure 5: Atomic density for grain growth at times $t=250, 500, 1000, 2000, 3000, 4000$ (from left to right, top to bottom). The domain is $(0,201)^2$, the mesh consists of $402 \times 402$ elements, and $\Delta t = 1$. The colors are scaled from $\phi = -0.47$ to $\phi = 0.63$. See \ref{['ss:grain_grows']} for more details.
  • ...and 2 more figures

Theorems & Definitions (14)

  • Lemma 3.1: Energy stability of the semi-discrete problem
  • proof
  • Lemma 3.2: Fully-discrete energy stability
  • proof
  • Theorem 4.1: Brouwer's fixed-point theorem
  • Lemma 4.2
  • Lemma 4.3
  • proof
  • Theorem 4.4: Existence of a solution
  • proof
  • ...and 4 more