### Abstract

We present a finite element discretization of a non-linear diffusion equation used in the field of critical phenomena and, more recently, in the context of dynamic density functional theory. The discretized equation preserves the structure of the continuum equation. Specifically, it conserves the total number of particles and fulfills an H-theorem as the original partial differential equation. The discretization proposed suggests a particular definition of the discrete hydrodynamic variables in microscopic terms. These variables are then used to obtain, with the theory of coarse-graining, their dynamic equations for both averages and fluctuations. The hydrodynamic variables defined in this way lead to microscopically derived hydrodynamic equations that have a natural interpretation in terms of discretization of continuum equations. Also, the theory of coarse-graining allows to discuss the introduction of thermal fluctuations in a physically sensible way. The methodology proposed for the introduction of thermal fluctuations in finite element methods is general and valid for both regular and irregular grids in arbitrary dimensions. We focus here on simulations of the Ginzburg-Landau free energy functional using both regular and irregular 1D grids. Convergence of the numerical results is obtained for the static and dynamic structure factors as the resolution of the grid is increased.

Original language | English (US) |
---|---|

Article number | 094115 |

Journal | Journal of Chemical Physics |

Volume | 142 |

Issue number | 9 |

DOIs | |

State | Published - Mar 7 2015 |

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### ASJC Scopus subject areas

- Physics and Astronomy(all)
- Physical and Theoretical Chemistry

### Cite this

*Journal of Chemical Physics*,

*142*(9), [094115]. https://doi.org/10.1063/1.4913746

**Finite element discretization of non-linear diffusion equations with thermal fluctuations.** / De La Torre, J. A.; Español, Pep; Donev, Aleksandar.

Research output: Contribution to journal › Article

*Journal of Chemical Physics*, vol. 142, no. 9, 094115. https://doi.org/10.1063/1.4913746

}

TY - JOUR

T1 - Finite element discretization of non-linear diffusion equations with thermal fluctuations

AU - De La Torre, J. A.

AU - Español, Pep

AU - Donev, Aleksandar

PY - 2015/3/7

Y1 - 2015/3/7

N2 - We present a finite element discretization of a non-linear diffusion equation used in the field of critical phenomena and, more recently, in the context of dynamic density functional theory. The discretized equation preserves the structure of the continuum equation. Specifically, it conserves the total number of particles and fulfills an H-theorem as the original partial differential equation. The discretization proposed suggests a particular definition of the discrete hydrodynamic variables in microscopic terms. These variables are then used to obtain, with the theory of coarse-graining, their dynamic equations for both averages and fluctuations. The hydrodynamic variables defined in this way lead to microscopically derived hydrodynamic equations that have a natural interpretation in terms of discretization of continuum equations. Also, the theory of coarse-graining allows to discuss the introduction of thermal fluctuations in a physically sensible way. The methodology proposed for the introduction of thermal fluctuations in finite element methods is general and valid for both regular and irregular grids in arbitrary dimensions. We focus here on simulations of the Ginzburg-Landau free energy functional using both regular and irregular 1D grids. Convergence of the numerical results is obtained for the static and dynamic structure factors as the resolution of the grid is increased.

AB - We present a finite element discretization of a non-linear diffusion equation used in the field of critical phenomena and, more recently, in the context of dynamic density functional theory. The discretized equation preserves the structure of the continuum equation. Specifically, it conserves the total number of particles and fulfills an H-theorem as the original partial differential equation. The discretization proposed suggests a particular definition of the discrete hydrodynamic variables in microscopic terms. These variables are then used to obtain, with the theory of coarse-graining, their dynamic equations for both averages and fluctuations. The hydrodynamic variables defined in this way lead to microscopically derived hydrodynamic equations that have a natural interpretation in terms of discretization of continuum equations. Also, the theory of coarse-graining allows to discuss the introduction of thermal fluctuations in a physically sensible way. The methodology proposed for the introduction of thermal fluctuations in finite element methods is general and valid for both regular and irregular grids in arbitrary dimensions. We focus here on simulations of the Ginzburg-Landau free energy functional using both regular and irregular 1D grids. Convergence of the numerical results is obtained for the static and dynamic structure factors as the resolution of the grid is increased.

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U2 - 10.1063/1.4913746

DO - 10.1063/1.4913746

M3 - Article

VL - 142

JO - Journal of Chemical Physics

JF - Journal of Chemical Physics

SN - 0021-9606

IS - 9

M1 - 094115

ER -