Finite Element Approximation of Partial Differential Equations

偏微分方程的有限元逼近

• 批准号: 0609755
• 负责人: Richard Falk
• 金额: $ 18.99万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0609755&HistoricalAwards=false
• 关键词: Finite; Element; Approximation; Partial; Differential

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.

建议研究的第一个领域涉及研究兼容的离散方案偏微分方程，一种方法，试图产生数值逼近，继承或模仿的基本性质的偏微分方程，如守恒和对称性。 几个问题，建模现象非弹性和流体，将从这个角度进行研究。新方法是基于分段多项式精确弹性序列的构造，它与稳定混合有限元格式的发展密切相关。 在这方面的其他工作包括一个新的方法来建设的层次基地的标量和向量值的有限元空间在任意空间维度，和确切的序列性质的矩形和四边形有限元及其应用的稳定性的混合有限元逼近。 第二个研究领域是定义在不规则六面体单元上的几种类型的有限元空间的逼近性质，这些单元是由参考立方体的三线性映射得到的。这样的空间被用来近似三维向量函数，并自然地出现在许多应用中，包括麦克斯韦方程的近似和二阶椭圆方程的混合和最小二乘有限元方法的使用。虽然人们常常隐含地假设，已知的近似结果正六面体扩展到这些空间，事实上，这是不是这样的。 本文的研究目的是精确地确定最佳逼近阶所需的条件，并构造具有这一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的间断Galerkin方法的收敛速度。 虽然最优阶收敛率在实践中经常看到，理论保证这样的速度只为均匀网格，而较低的速度是已知的最好的可能在专门构造的网格。 提出的研究是分类类型的meshesfor的最佳收敛率是achieved.Mathematical建模的物理和生物过程使用partialdifferential方程已成为标准的方法研究主机ofimportant问题.这样的模型捕捉在一个简洁和精确的方式thefundamental功能的过程被建模。不幸的是，由此产生的方程很少有解，可以用简单的数学公式表示。 因此，发展可靠和有效的数值逼近方案是必要的，使这种方法成为apractical方法，是中央的进步，在许多领域的科学和工程。 这一发展的一部分涉及到对数值方法的理论基础的研究。 这样的调查可以导致更好地了解现有的方法和发展的新方法与理想的性能。因此，这种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。 本计画主要研究弹性力学与流体流动现象之近似方程式之数值方法。 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，以便更准确地捕捉被建模的基础过程的基本特征。