A finite element method for stochastic diffusion equations using fluctuating hydrodynamics

Abstract

We present a finite element approach for diffusion problems with thermal fluctuations based on a fluctuating hydrodynamics model. The governing transport equations are stochastic partial differential equations with a fluctuating forcing term. We propose a discrete formulation of the stochastic forcing term that has the correct covariance matrix up to a standard discretization error. Furthermore, to obtain a numerical solution with spatial correlations that converge to those of the continuum equation, we derive a linear mapping to transform the finite element solution into an equivalent discrete solution that is free from the artificial correlations introduced by the spatial discretization. The method is validated by applying it to two diffusion problems: a second-order diffusion equation and a fourth-order diffusion equation. The theoretical (continuum) solution to the first case presents spatially decorrelated fluctuations, while the second case presents fluctuations correlated over a finite length. In both cases, the numerical solution presents a structure factor that approximates well the continuum one.

Figures (16)

• Figure 1: Left: Structure factor of the discrete solution; theoretical structure factor for the discrete equation (solid line) and computed with $N_n=51$ (dashed) and $N_n=101$ (dotted). Right: error ${e_{FE}}$ in the computed discrete structure factor vs the number of elements ($\circ$). All computations have $p1$-elements, $\Delta t=1.0e-4$, $N_t=1.0e6$, $u_{0}=10000$. It can be seen that the solution obtained from the FE computation converges to the theoretical solution of the discretized equation as the number of elements increases.
• Figure 2: Structure factor of the discrete solution as a function of $k L$; computed with $N_n=51$ (dashed), $N_n=101$ (dotted) and $N_n=201$ (dash-dotted). All computations with $p1$-elements, $\Delta t=1.0e-4$, $N_t=1.0e6$, $u_{0}=10000$.
• Figure 3: Structure factor of the discrete solution computed with $p1$-elements (solid line) and $p2$-elements (dash-dot). The computations have been run with $N_{n}=101$, $\Delta t=1.0e-4$, $N_t=1.0e6$, $u_{0}=10000$.
• Figure 4: Structure factor of the discrete solution obtained with $p2$-elements, computed with $N_{n}=51$ (dashed) and $N_{n}=101$ (dotted). The computations have been run with $\Delta t=1.0e-4$, $N_t=1.0e6$, $u_{0}=10000$.
• Figure 5: Structure factor of the discrete solution with $p1$-elements, computed with $N_n=51$ (dashed) and $N_n=101$ (dotted). Left: fully implicit time integration ($\alpha=0$). Right: fully explicit time integration ($\alpha=1$). All computations with $\Delta t=1.0e-5$, $N_t=1.0e6$, $u_{0}=10000$. As a reference, numerical results obtained with $\alpha=1/2$ are also displayed (solid line), which are equivalent to the case with $\beta=0$.