We consider the Fokker–Planck equation (FPE) for the orientation probability density of fiber suspensions. Using the continuous Galerkin method, we express the numerical solution in terms of Lagrange basis functions that are associated with N nodes of a computational mesh for a domain in the 3D physical space and M nodes of a mesh for the surface of a unit sphere representing the configuration space. The NM time-dependent unknowns of our finite element approximations are probabilities corresponding to discrete space locations and orientation angles. The framework of alternating-direction methods enables us to update the numerical solution in parallel by solving N evolution equations on the sphere and M three-dimensional advection equations in each (pseudo-)time step. To ensure positivity preservation as well as the normalization property of the probability density, we perform algebraic flux correction for each equation and synchronize the correction factors corresponding to different orientation angles. The velocity field for the spatial advection step is obtained using a Schur complement method to solve a generalized system of the incompressible Navier–Stokes equations (NSE). Fiber-induced subgrid-scale effects are taken into account using an effective stress tensor that depends on the second- and fourth-order moments of the orientation density function. Numerical studies are performed for individual subproblems and for the coupled FPE-NSE system.

中文翻译：

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使用符合物理的有限元方法对 Fokker-Planck 方程进行高效数值求解

我们考虑福克-普朗克方程（FPE）来计算纤维悬浮液的取向概率密度。使用连续伽辽金方法，我们用与以下相关的拉格朗日基函数来表达数值解：氮3D 物理空间中域的计算网格的节点，以及中号表示配置空间的单位球体表面的网格节点。这纳米有限元近似的时间相关未知数是与离散空间位置和方向角相对应的概率。交替方向方法的框架使我们能够通过求解并行更新数值解氮球体上的演化方程和中号每个（伪）时间步长的三维平流方程。为了确保正性保留以及概率密度的归一化特性，我们对每个方程进行代数通量校正，并同步不同方向角对应的校正因子。空间平流步骤的速度场是使用 Schur 补法求解不可压缩纳维-斯托克斯方程 (NSE) 的广义系统获得的。使用取决于取向密度函数的二阶和四阶矩的有效应力张量考虑纤维引起的亚网格尺度效应。对单个子问题和耦合的 FPE-NSE 系统进行数值研究。