论文标题
动力学fokker-Planck方程的稳定且高效
Stable and efficient Petrov-Galerkin methods for a kinetic Fokker-Planck equation
论文作者
论文摘要
我们提出了一个稳定的Petrov-Galerkin离散化，以一种方式构建的动力学Fokker-Planck方程，即可以直接从变化公式中推断出均匀的INF-SUP稳定性。受抛物线方程的适应性结果的启发，我们通过稳定的试验和测试功能对Fokker-Planck双线性形式的双inf-SUP常数得出了一个下限。通过将动力学转运操作员和逆速度拉普拉斯 - 贝特拉米操作员应用于给定的测试功能，可以构建这种对的试验功能。对于Petrov-Galerkin投影，我们选择一个任意的离散测试空间，然后使用相同的运输和逆Laplace-Beltrami运算符定义离散试验空间。结果，空间复制了连续水平的稳定对，我们获得了一个良好的数值方法，其具有离散的INF-SUP常数与连续问题的INF-SUP常数相同，独立于网格大小。我们展示了如何通过低维椭圆问题有效地计算特定基础功能，并通过数值实验确认该方法的实用性和性能。
We propose a stable Petrov-Galerkin discretization of a kinetic Fokker-Planck equation constructed in such a way that uniform inf-sup stability can be inferred directly from the variational formulation. Inspired by well-posedness results for parabolic equations, we derive a lower bound for the dual inf-sup constant of the Fokker-Planck bilinear form by means of stable pairs of trial and test functions. The trial function of such a pair is constructed by applying the kinetic transport operator and the inverse velocity Laplace-Beltrami operator to a given test function. For the Petrov-Galerkin projection we choose an arbitrary discrete test space and then define the discrete trial space using the same application of transport and inverse Laplace-Beltrami operator. As a result, the spaces replicate the stable pairs of the continuous level and we obtain a well-posed numerical method with a discrete inf-sup constant identical to the inf-sup constant of the continuous problem independently of the mesh size. We show how the specific basis functions can be efficiently computed by low-dimensional elliptic problems, and confirm the practicability and performance of the method with numerical experiments.
