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

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. Guided by the Theory of Coarse-Graining, we discuss the inclusion of thermal fluctuations in the non-linear diffusion equation. This sheds light on the meaning of such a fluctuating hydrodynamics equation and to the limitations of the approximations involved. 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.