Finite Element Approximation of Partial Differential Equations

第一个研究领域涉及偏微分方程相容离散化的研究，即继承或模仿偏微分方程基本性质的数值逼近格式。所提出的研究试图利用这样一种想法，即导致偏微分方程边值问题适定性的一些相同的概念可以被用来为它们的逼近开发相容和稳定的有限元格式。我们的计划是利用这一思想来发展一种更简单、更系统的有限元分析方法，在以往工作的基础上，设计出适用于各种应用问题的新的、更好的数值逼近方案。提出的第二个项目是对P.I.最近开发的一种新的数值近似格式的进一步发展，该格式用于模拟平行板在介质器件上的电润湿。电润湿是一种基于电和表面张力现象之间的关系的效应，它允许通过使用施加的电压来控制液-气界面的形状和运动。当施加电压时，液体表面会改变形状，以使表面张力能和电能之和最小。建议的研究是首先从计算上研究新方法，看看结果与实验结果如何比较，然后建立方法的稳定性并推导出误差估计。该方法的一些新特点是，改变的边界位置现在是公式中的一个变量，未知速度可以用标准的简单有限元空间来近似。最后一个项目是设计新的更简单的四边形和六面体有限元，用于偏微分方程组的混合有限元逼近。一个主要问题是，当这些元素以通常的方式定义时，不会保留与正方形和立方体上定义的元素相同的近似属性。使用偏微分方程对物理和生物过程进行数学建模已成为研究一系列重要问题的标准方法。这样的模型以一种简洁和精确的方式捕捉到了被建模的过程的基本特征。不幸的是，所得到的方程很少有可以用简单的数学公式表示的解。因此，发展可靠和有效的数值逼近方案是必要的，以使这种方法成为一种实用的方法，并是许多科学和工程领域的进步的核心。这一发展的一部分涉及对数值方法的理论基础的研究。这样的研究可以使人们更好地理解现有的方法，并开发出具有理想性能的新方法。因此，这样的研究有可能提高科学家和工程师进行的基本计算机模拟的准确性，甚至使其成为可能。该项目主要研究在各种应用中模拟现象的近似方程的数值方法。一个中心主题是开发近似方案，以保留数学模型的一些关键属性的离散版本，以便更准确地捕获正在建模的基本过程的基本特征。

The first area of proposed research involves the study of compatible discretizations of partial differential equations, i.e., numerical approximation schemes that inherit or mimic fundamental properties of a partial differential equation. The proposed research seeks to capitalize on the idea that some of the same concepts that lead to the well-posedness ofboundary value problems for partial differential equations can be used to develop compatible and stable finite element schemes for their approximation. The plan is to use this idea to develop a simpler and more systematic analysis of finite element methods, which based on past work, should then lead to the design of new and better numerical approximation schemes for a variety of applied problems. The second project proposed is the further development of a new numerical approximation scheme recently developed by the P.I. for the simulation of a parallel-plate Electrowetting on Dielectric Device. Electrowetting is an effect, based on a relationship between electrical and surface tension phenomena, that allows for control of the shape and motion of a liquid-gas interface, through the use of an applied voltage. The liquid surface changes shapes when a voltage is applied in order to minimize the sum of the surface tension energy and electrical energy. The proposed research is to first investigate the new approach computationally to see how the results compare with experiments, and then to establish stability of the method and derive error estimates. Some novel features of the method are that the changing boundary position is now a variable in the formulation and the unknown velocity can be approximated by standard simple finite element spaces. The final project is the design of new simpler quadrilateral and hexahedral finite elements for use in the mixed finite element approximation of partial differential equations. A main issue is that such elements, when defined in the usual way, do not retain the same approximation properties as those defined on squares and cubes. Mathematical modeling of physical and biological processes using partial differential equations has become the standard method of studying a host of important problems. Such models capture in a concise and precise way the fundamental features of the process being modeled. Unfortunately, the resulting equations rarely have solutions that can be expressed by simple mathematical formulas. Hence, the development of reliable and efficient numerical approximation schemes are necessary to make this method into a practical approach and is central to progress in many areas of science and engineering. Part of this development involves the investigation of the theoretical underpinnings of numerical methods. Such investigation can lead to a greater understanding of existing methods and to the development of new methods with desirable properties. Thus, such study has the potential to improve the accuracy of, or even make possible, essential computer simulations performed by scientists and engineers. This project is concerned with the study of numerical methods for approximating equations modeling phenomena in a variety of applications. One central theme is to develop approximation schemes that preserve discrete versions of some of the key properties of the mathematical model, in order to more accurately capture the fundamental features of the underlying process being modeled.