New techniques in characteristic finite element methods for flow problems

这项提案是利用2009年《美国复苏和再投资法案》(公法111-5)提供的资金授予的。所提出的研究重点是设计、分析和实施流动问题的准确和高效的数值方法。众所周知，流动问题中的不可压缩性和对流占优势给数值模拟器带来了巨大的挑战。在这个研究项目中，PI和他的合作者将开发基于局部无散度(LDF)有限元和特征跟踪的数值格式来克服上述两个主要困难。LDF有限元在h-自适应(由于基函数与单元几何无关)和p-自适应(由于基函数的层次结构)方面都具有很强的潜力。离散Helmholtz分解和杂交将被用来解耦得到的离散线性系统，并降低计算成本。高阶特征方法被开发用于提高精度的长期模拟。这些新技术将被结合到不连续Galerkin框架中，以开发快速而稳健的Stokes方程、Navier-Stokes方程、对流-扩散-反应方程和磁流体动力学方程的解算器。拟议的研究的成功将直接影响一大类数值模拟器的效率和稳健性，这些问题包括地下水污染修复、细胞内蛋白质运输、核废物处理、石油回收、天气预报和野火控制。该项目开发的新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，让他们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中的各种应用。

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.