Computational Methods in Science and Engineering

Efficient Numerical Methods for Computing the Stationary States of Phase Field Crystal Models


Finding the stationary states of a free energy functional is an important problem in phase field crystal (PFC) models. Many efforts have been devoted to designing numerical schemes with energy dissipation and mass conservation properties. However, most existing approaches are time-consuming due to the requirement of small effective step sizes. In this paper, we discretize the energy functional and propose efficient numerical algorithms for solving the constrained nonconvex minimization problem. A class of gradient-based approaches, which are the so-called adaptive accelerated Bregman proximal gradient (AA-BPG) methods, is proposed, and the convergence property is established without the global Lipschitz constant requirements. A practical Newton method is also designed to further accelerate the local convergence with convergence guarantee. One key feature of our algorithms is that the energy dissipation and mass conservation properties hold during the iteration process. Moreover, we develop a hybrid acceleration framework to accelerate the AA-BPG methods and most of the existing approaches through coupling with the practical Newton method. Extensive numerical experiments, including two three-dimensional periodic crystals in the Landau--Brazovskii (LB) model and a two-dimensional quasicrystal in the Lifshitz--Petrich (LP) model, demonstrate that our approaches have adaptive step sizes which lead to a significant acceleration over many existing methods when computing complex structures.


  1. phase field crystal models
  2. stationary states
  3. adaptive accelerated Bregman proximal gradient methods
  4. preconditioned conjugate gradient method
  5. hybrid acceleration framework

MSC codes

  1. 35J60
  2. 35Q74
  3. 65N35

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Information & Authors


Published In

cover image SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Pages: B1350 - B1377
ISSN (online): 1095-7197


Submitted: 24 February 2020
Accepted: 7 August 2020
Published online: 9 November 2020


  1. phase field crystal models
  2. stationary states
  3. adaptive accelerated Bregman proximal gradient methods
  4. preconditioned conjugate gradient method
  5. hybrid acceleration framework

MSC codes

  1. 35J60
  2. 35Q74
  3. 65N35



Funding Information

Hunan Science Foundation of China : 2018JJ2376
Chinese Academy of Sciences Key Project : 19A500
Education Department of Hunan Province
National Social Science Fund Youth Project : 18B057
National Natural Science Foundation of China : 11771368, 11901338

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