Finite Element Approximation of Partial Differential Equations

研究的第一个领域涉及偏微分方程的相容离散化方案的研究，这是一种试图产生继承或模仿偏微分方程基本性质的数值近似的方法，如守恒性和对称性。从这个角度出发，我们将研究几个问题，即建模现象、弹性和流体。所采用的新方法是基于分段多项式精确序列的构造，而分段多项式精确序列的构造与稳定有限元格式的发展密切相关。这一领域的其他工作包括构造任意空间维标量和矢量值有限元空间的层次基的新方法，矩形和四边形有限元的精确序列性质及其在混合有限元逼近稳定性中的应用。第二个研究领域是定义在由参考立方体的三线性映射得到的正六面体单元上的几类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以及二阶椭圆型方程的混合有限元和最小二乘有限元方法的使用。虽然人们经常隐含地假设已知的正六面体的近似结果推广到这些空间，但事实并非如此。本研究的目的是精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。第三个研究领域是研究线性双曲问题间断Galerkin方法的收敛速度。虽然最优阶收敛速度在实际中经常出现，但理论上只对均匀网格保证这样的收敛速度，而较低的收敛速度在特殊构造的网格上是最好的。提出的研究是对获得最优收敛速度的网格类型进行分类。使用偏微分方程组对物理和生物过程进行数学建模已成为研究许多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，这些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格式对于使该方法成为实际方法是必要的，并且对于许多科学和工程领域的进步是至关重要的。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。本课题致力于研究用于模拟弹性和流体流动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，保留数学模型的一些基本属性的离散版本，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 compatiblediscretization schemes for partial differential equations, an approach thatattempts to produce numerical approximations that inherit or mimicfundamental properties of a partial differential equation, such asconservation and symmetries. Several problems, modeling phenomena inelasticity and fluids, will be studied from this point of view. The newapproach taken is based on the construction of piecewise polynomial exactelasticity sequences, which are closely related to the development of stablemixed finite element schemes. Other work in this area includes a new approach to the construction of hierarchical bases for scalar and vector-valued finite element spaces in arbitrary space dimensions, and exact sequence properties of rectangular and quadrilateral finite elements and their applications to thestability of mixed finite element approximation. The second area of study is the approximation properties of several types of finite element spaces defined on irregular hexahedral elements obtained by trilinear mappings from areference cube. Such spaces are used to approximate three-dimensional vectorfunctions and arise naturally in many applications, including the approximation of Maxwell's equations and the use of mixed and least squares finite elementmethods for second order elliptic equations. Although it is often implicitlyassumed that approximation results known for regular hexahedrons extend tothese spaces, in fact this is not the case. The research is to determineprecisely what is needed for optimal order approximation and constructfamilies of finite element spaces that have this property. The third area of research is to study convergence rates for discontinuous Galerkin methods for linear hyperbolic problems. Although optimal order convergence rates areoften seen in practice, the theory guarantees such rates only for uniformmeshes, while a lower rate is known to be the best possible on speciallyconstructed meshes. The proposed research is to classify the type of meshesfor which the optimal convergence rate is achieved.Mathematical modeling of physical and biological processes using partialdifferential equations has become the standard method of studying a host ofimportant problems. Such models capture in a concise and precise way thefundamental features of the process being modeled. Unfortunately, theresulting equations rarely have solutions that can be expressed by simplemathematical formulas. Hence, the development of reliable and efficientnumerical approximation schemes are necessary to make this method into apractical approach and is central to progress in many areas of science andengineering. Part of this development involves the investigation of thetheoretical underpinnings of numerical methods. Such investigation can leadto a greater understanding of existing methods and to the development of newmethods with desirable properties. Thus, such study has the potential toimprove the accuracy of, or even make possible, essential computer simulations performed by scientists and engineers. This project is concerned with thestudy of numerical methods for approximating equations modeling phenomena inelasticity and fluid flow. One central theme is to develop approximationschemes that preserve discrete versions of some of the fundamental properties of the mathematical model, in order to more accurately capture the fundamental features of the underlying process being modeled.