Finite Element Approximation of Partial Differential Equations

第一个研究领域是定义在由参考立方体的三线性映射得到的不规则六面体单元上的几种有限元空间的逼近性质。这种空间被用来逼近三维向量函数，并自然地出现在许多应用中，包括麦克斯韦方程的逼近和二阶椭圆型方程的混合最小二乘有限元方法。这项研究的目的是准确地确定最优阶逼近所需的条件，并构造具有这种性质的有限元空间族。第二个研究领域是对流扩散问题的间断Galerkin方法的有限元逼近。我们的目的是得到新的局部误差估计，以便了解这类有希望的逼近格式中的哪种方法对扩散占优和对流占优的二阶偏微分方程组都适用。第三个研究领域是利用目前成熟的Reissner-Mindlin板模型(用于研究薄板在外力作用下的弯曲)的近似理论，作为发展使用有限元方法近似弹性壳的新方法的基础。当采用标准的有限元近似格式时，板模型和壳模型在更大程度上都存在“锁定”问题，导致对薄板和壳的近似较差。最后一个研究领域涉及爱因斯坦方程的有效数值方法的设计，该方法用于数值模拟黑洞碰撞等大规模天文事件的引力辐射发射。所采取的方法将是使用具有一些相同特征的更简单的模型问题来理解为什么爱因斯坦方程的标准数值方法失败，并帮助设计克服这些问题的方法。使用偏微分方程组对物理和生物过程进行数学建模已经成为研究一系列重要科学问题的标准方法。由于通常不可能精确地求解此类方程，可在计算机上实现的可靠和有效的数值逼近格式的发展使之成为一种实用的方法，并对许多科学和工程领域的进步起着核心作用。该项目研究各种应用问题的数学模型的“有限元”型近似格式。其中包括同时存在对流和扩散的气体和流体流动，用于模拟受外加电流的物体中的电场和磁场的麦克斯韦方程，在外部载荷下薄结构(例如，Aroof)的弯曲，以及用于模拟大规模天文事件(如黑洞碰撞)产生的引力辐射发射的爱因斯坦方程。这项工作有望带来新的和改进的数值方法，供科学家和工程师在应用计算中使用。

The first area of study is the approximation properties of several types offinite element spaces defined on irregular hexahedral elements obtained bytrilinear mappings from a reference cube. Such spaces are used to approximate three-dimensional vector functions and arise naturally in many applications,including the approximation of Maxwell's equations and the use of mixed andleast squares finite element methods for second order elliptic equations. The research is to determine precisely what is needed for optimal orderapproximation and construct families of finite element spaces that have thisproperty. The second area of study is the finite element approximation bydiscontinuous Galerkin methods of convection-diffusion problems. The aim isto derive new local error estimates in order to understand which methods ofthis promising class of approximation schemes work well both fordiffusion-dominated and convection-dominated second order partial differential equations. The third area of research is to use the now well-developed theory for the approximation of the Reissner-Mindlin plate model (used to study thebending of a thin plate under external loads) as a basis for developing newapproaches to the use of finite element methods for the approximation ofelastic shells. Both the plate model and to a greater extent the shell model suffer from the problem of "locking'' when standard finite elementapproximation schemes are applied, causing poor approximations for thin plates and shells. The final area of research involves the design of effectivenumerical methods for the Einstein equations, used to numerically simulate the emission of gravitation radiation from massive astronomical events such asblack hole collisions. The approach taken will be to use simpler modelproblems with some of the same features to understand why standard numericalmethods for the Einstein equations fail and to help design methods thatovercome these problems.The mathematical modeling of physical and biological processes using partialdifferential equations has become the standard method of studying a host ofimportant scientific problems. Since it is usually not possible to solve such equations exactly, the development of reliable and efficient numericalapproximation schemes, which can be implemented on computers, makes this into a practical approach and is central to progress in many areas of science andengineering. This project studies "finite element" type approximation schemes for mathematical models of a variety of applied problems. These include flows of gases and fluids in which both convection and diffusion are present,Maxwell's equations for the modeling of the electric and magnetic fields in a body subject to an applied current, the bending of thin structures (e.g., aroof) under external loads, and Einstein's equations for the simulation of the emission of gravitation radiation from massive astronomical events such asblack hole collisions. This work is expected to lead to new and improvednumerical methods for use by scientists and engineers in applied computations.