A Scalable High-Order Discontinuous Finite Element Framework for Partial Differential Equations: with Application to Geophysical Fluid Flows

偏微分方程（PDE）在工程和科学中普遍存在，其数值解对于理解复杂的自然，工程和社会系统至关重要。虽然在过去的几十年里，在偏微分方程的理论和计算算法方面都取得了巨大的进步，但针对复杂、耦合和多物理场系统的可扩展数值方法仍然具有挑战性。这将成为未来艾级系统的主要障碍，除非数学方法和计算算法发生重大变化。然而，没有太多的工作已经做了计算数学的这一关键组成部分。因此，迫切需要开发先进的数学离散化，使科学应用能够利用极端规模计算的潜力，以继续科学发现的步伐并促进科学进步。 PI将开发一个高阶不连续有限元（FE）框架，包括弱Galerkin和杂交不连续Galerkin方法，以及用于地球物理流体动力学应用的可扩展求解器。特别是，PI将设计和严格分析一个抽象的高阶不连续有限元框架的一个大类的偏微分方程，包括椭圆，抛物线和双曲型。PI还将开发、分析和实现可扩展的求解器。将开发流体静力学和非流体静力学模型，并作为开发的试验台。该项目中开发的先进计算和数学方法将潜在地影响地球物理流体动力学的计算。该项目将为未来高分辨率地球系统模型的动力学核心提供有竞争力的极端尺度离散化。

Partial differential equations (PDEs) are pervasive in engineering and science, and their numerical solutions are of paramount importance in understanding complex, natural, engineered, and societal systems. Though the past decades have seen tremendous advances in both theories and computational algorithms for PDEs, scalable numerical methods for complex, coupled, and multiphysics systems that fully exploit the extreme-scale computing systems remain challenging. This becomes the major impediment for future exascale systems unless dramatic changes in mathematical methods and computational algorithms take place. However, not much work has been done for this critical component of computational mathematics. Thus, there is a critical need to develop advanced mathematical discretizations that can enable scientific applications to harness the potential of extreme-scale computing in order to continue the pace of scientific discoveries and to promote the progress of science. The PI will develop a high-order discontinuous finite element (FE) framework, including the weak Galerkin and the hybridized discontinuous Galerkin methods, and its scalable solver for geophysical fluid dynamic applications. In particular, the PI will design and rigorously analyze an abstract high-order discontinuous FE framework for a large class of PDEs including elliptic, parabolic, and hyperbolic types. The PI will also develop, analyze, and implement scalable solvers. Both hydrostatic and non-hydrostatic models will be developed and served as test beds for the developments. The advanced computational and mathematical methods developed in this project will potentially impact the computation of geophysical fluid dynamics. The project will contribute a competitive extreme-scale discretization for the dynamical cores of future high-resolution earth system models.