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.

翻译：本文提出一种基于涨落流体力学模型的有限元方法，用于处理含热涨落的扩散问题。控制输运方程为带随机强迫项的随机偏微分方程。我们提出随机强迫项的离散表达式，其在标准离散误差范围内具有正确的协方差矩阵。此外，为获得与连续方程具有相同空间相关性的数值解，我们推导出线性映射，将有限元解转化为等效离散解，从而消除空间离散引入的人为相关性。通过将方法应用于两个扩散问题（二阶扩散方程和四阶扩散方程）进行验证。第一个问题的理论（连续）解呈现空间去相关涨落，而第二个问题的涨落则具有有限长度相关性。两种情况下，数值解的结构因子均与连续解吻合良好。