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

本项目由2009年美国经济复苏和再投资法案（公法111-5）资助，主要研究方向为开发和分析结构力学和流体流动问题的不连续Galerkin （DG）方法。特别地，P.I.将分析板弯曲问题，弹性方程和对流扩散方程的杂交不连续伽辽金（HDG）方法。HDG方法的一个优点是，它们可以以最优的方式近似所有感兴趣的变量，同时对所有变量使用等阶近似。更重要的是，许多全局自由度可以通过使用拉格朗日乘子来消除，使最终的线性系统比标准DG方法中产生的线性系统更小。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.希望发展使用非多项式基函数的高阶DG方法。期末项目将回答有限元素的理论问题。本文将证明在一般凸多面体域上应用于Stokes问题的有限元方法的点误差估计。然后，pi将证明高阶流线扩散方法在层适应网格上的误差估计。数值模拟在现代工程中起着核心作用。例如，它们在飞机、汽车和石油平台的设计中起着至关重要的作用。它们允许工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，非常有效和可靠的数值方法已经发展了多年。然而，为了应对新的计算挑战，研究人员正在努力改进现有算法并开发新的竞争性算法。在这个项目中，P.I.将致力于开发一种新的，有前途的数值方法，称为杂交不连续伽辽金方法。为了更深入地了解这些数值方法和相关方法，P.I.也将研究这些方法的数学方面。

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.