Hybridizable Discontinuous Galerkin Methods for Partial Differential Equations and Theoretical Questions in Finite Elements

偏微分方程与有限元理论问题的混合间断伽辽金法

• 批准号: 0914596
• 负责人: Johnny Guzman
• 金额: $ 18.98万
• 依托单位: Brown University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-08-01 至 2013-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0914596&HistoricalAwards=false
• 关键词: Hybridizable; Discontinuous; Galerkin; Methods; Partial

This proposal is awarded using funds made available by the American Recovery and Reinvestment Act of 2009 (Public Law 111-5), The main part of this project will focus on developing and analyzing discontinuous Galerkin (DG) methods for problems arising in structural mechanics and fluid flow. In particular, the P.I. will analyze hybridizable discontinuous Galerkin (HDG) methods for plate bending problems, elasticity equations and convection-diffusion equations. One advantage of HDG methods is that they can approximate all the variables of interest in an optimal way while using equal-order approximations for all the variables. More importantly, many of the global degrees of freedom can be eliminated by the use of Lagrange multipliers, making the final linear system smaller than linear systems arising in standard DG methods. Another component of the project is the investigation of DG methods for multiscale problems. The P.I. hopes to develop higher-order DG methods using non-polynomial basis functions. A final project will be answering theoretical questions in finite elements. The P.I. will prove pointwise error estimates for finite element methods applied to the Stokes problem on general convex polyhedral domains. Then, the P.I. will prove error estimates for higher-order streamline diffusion methods on layer-adapted meshes. Numerical simulations play a central role in modern engineering. For example, they are crucial in the design of airplanes, automobiles, and oil platforms, to name a few. They allow industries to test structures using computers without ever building an actual physical model. One of the reasons this is possible is that very efficient and reliable numerical methods have been developed over the years. However, to meet new computational challenges, researchers are working on improving existing algorithms and on the development of new competitive ones. In this project, the P.I. will work on developing a new, promising family of numerical methods called hybridizable discontinuous Galerkin methods. In order to gain a deeper understanding of these numerical methods and related ones, the P.I. will also investigate mathematical aspects of such methods.

该提案是使用2009年《美国恢复和再投资法》（公法111-5）提供的资金授予​​的，该项目的主要部分将着重于开发和分析不连续的盖尔金（DG）方法，用于在结构机械和流体流中引起的问题。特别是P.I.将分析用于板弯曲问题，弹性方程和对流扩散方程的杂交不连续的盖金（HDG）方法。 HDG方法的一个优点是，它们可以以最佳方式近似所有感兴趣的变量，同时使用所有变量的等序近似值。更重要的是，可以通过使用拉格朗日乘数来消除许多全球自由度，从而使最终线性系统小于标准DG方法中产生的线性系统。 该项目的另一个组成部分是研究多尺度问题的DG方法。 P.I.希望使用非多功能基函数开发高阶DG方法。 最终项目将在有限元素中回答理论问题。 P.I.将证明将有限元方法应用于一般凸多面体域上的Stokes问题的有限元方法。 然后，P.I.将证明对层适应的网格上的高阶流线扩散方法进行错误估计。 数值模拟在现代工程中起着核心作用。 例如，它们对于飞机，汽车和石油平台的设计至关重要。 它们允许行业使用计算机测试结构，而无需建立实际的物理模型。这是可能的原因之一是多年来已经开发了非常有效且可靠的数值方法。 但是，为了应对新的计算挑战，研究人员正在努力改善现有算法和新竞争性算法的发展。在这个项目中，P.I.将致力于开发一个新的，有前途的数值方法家族，称为杂交不连续的Galerkin方法。为了更深入地了解这些数值方法和相关方法，P.I.还将研究此类方法的数学方面。