Spectral, Tensor and Domain Decomposition Methods for Fractional PDEs
Spectral, Tensor and Domain Decomposition Methods for Fractional PDEs
复制标题
DOI：
10.1515/cmam-2021-0118
复制
发表时间：
2022-02
影响因子：
1.3
通讯作者：
Tianyi Shi;Harbir Antil;D. Kouri
中科院分区：
文献类型：
--
作者：
Tianyi Shi;Harbir Antil;D. Kouri
摘要分数阶偏微分方程由于其非局部性和在捕捉界面间急剧变化时的灵活性，最近在物理学和成像科学中得到了广泛的应用。然而，这种非局部性使得它具有挑战性，设计有效的解决方案，这样的问题。在本文中，我们介绍了一种基于超球面多项式离散的Caffarelli-Silvestre扩展的谱方法来解决矩形和圆盘域上的偏微分方程。我们使用张量方程求解器解决离散化问题，从而可以解决高维偏微分方程。此外，我们介绍了串行和并行区域分解求解器。我们展示了我们的方法的数值性能上的三维分数椭圆PDE立方体以及分数PDE约束的优化问题的应用。
Abstract Fractional PDEs have recently found several geophysics and imaging science applications due to their nonlocal nature and their flexibility in capturing sharp transitions across interfaces. However, this nonlocality makes it challenging to design efficient solvers for such problems. In this paper, we introduce a spectral method based on an ultraspherical polynomial discretization of the Caffarelli–Silvestre extension to solve such PDEs on rectangular and disk domains. We solve the discretized problem using tensor equation solvers and thus can solve higher-dimensional PDEs. In addition, we introduce both serial and parallel domain decomposition solvers. We demonstrate the numerical performance of our methods on a 3D fractional elliptic PDE on a cube as well as an application to optimization problems with fractional PDE constraints.