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

Area of Science:

• Computational physics
• Mathematical modeling

Background:

• Non-linear diffusion equations are crucial in critical phenomena and dynamic density functional theory.
• Existing numerical methods may not fully preserve the physical properties of the continuum equations.

Purpose of the Study:

• To present a finite element discretization of a non-linear diffusion equation.
• To ensure the discretized equation conserves particle number and fulfills an H-theorem.
• To develop a method for introducing thermal fluctuations in finite element methods.

Main Methods:

• Finite element discretization of a non-linear diffusion equation.
• Definition of discrete hydrodynamic variables in microscopic terms.
• Application of coarse-graining theory to derive dynamic equations for averages and fluctuations.
• Simulation of the Ginzburg-Landau free energy functional on 1D grids.

Main Results:

• The finite element discretization preserves particle conservation and the H-theorem.
• Microscopically derived hydrodynamic equations with natural interpretations were obtained.
• A general methodology for introducing thermal fluctuations in finite element methods was established.
• Numerical simulations showed convergence for static and dynamic structure factors.

Conclusions:

• The proposed finite element method accurately discretizes non-linear diffusion equations.
• The method provides a physically sensible way to include thermal fluctuations.
• This approach is applicable to regular and irregular grids in arbitrary dimensions.