New techniques in characteristic finite element methods for flow problems

本提案使用2009年美国复苏和再投资法案（公法111-5）提供的资金进行奖励。研究的重点是设计、分析和实现精确、高效的流动问题数值方法。众所周知，流动问题中的不可压缩性和对流优势对数值模拟提出了重大挑战。在这个研究项目中，PI和他的合作者将开发基于局部无发散（LDF）有限元和特征跟踪的数值方案，以克服上述两个主要困难。LDF有限元具有很强的h-自适应性（由于基函数与元素几何的独立性）和p-自适应性（由于基函数的层次结构）。离散亥姆霍兹分解和杂化将被用于解耦离散线性系统并降低计算成本。高阶特性方法被开发用于长期模拟，提高了精度。这些新技术将被整合到不连续Galerkin框架中，以开发Stokes、Navier-Stokes、对流-扩散-反应和磁流体动力学方程的快速和健壮的求解器。该研究的成功将直接影响大量数值模拟器的效率和鲁棒性，这些数值模拟器可用于解决诸如地下水污染修复、细胞内蛋白质运输、核废料处理、石油开采、天气预报和野火控制等现实世界问题。本项目开发的新的数学方法和软件模块将为广泛的科学计算任务提供可靠的工具。该项目还将为研究生提供实践培训机会，以获得广泛的知识基础和复杂的技能，将数学和计算机模拟应用于科学和工程领域的各种应用。

This proposal is awarded using funds made available by the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The focus of the proposed research is to design, analyze, and implement accurate and efficient numerical methods for flow problems. As is well known, incompressibility and convection-dominance in flow problems pose significant challenges to numerical simulators. In this research project, the PI and his collaborators will develop numerical schemes based on the locally divergence-free (LDF) finite elements and characteristic tracking to overcome the aforementioned two major difficulties. The LDF finite elements have strong potentials for both h-adaptivity (due to the independence of the basis functions to element geometry) and p-adaptivity (due to the hierarchical structure of the basis functions). Discrete Helmholtz decomposition and hybridization will be utilized to decouple resulted discrete linear systems and reduce computational costs. Higher order characteristic methods are developed for long-term simulations with increased accuracy. These new techniques will be incorporated in the discontinuous Galerkin framework to develop fast and robust solvers for Stokes, Navier-Stokes, convection-diffusion-reaction, and magnetohydrodynamics equations.The success of the proposed research will have direct impact on the efficiency and robustness of a large class of numerical simulators for real world problems such as groundwater contamination remediation, intracellular protein trafficking, nuclear waste disposal, oil recovery, weather forecast, and wildfire control. The new mathematical methods and software modules developed in this project will provide reliable tools for a wide range of scientific computing tasks. This project will also provide hands-on training opportunities for graduate students to gain a wide knowledge base and sophisticated skills to apply mathematics and computer simulations to a variety of applications in sciences and engineering.