Finite Element Approximation of Partial Differential Equations

偏微分方程的有限元逼近

基本信息

  • 批准号:
    0308347
  • 负责人:
  • 金额:
    $ 17.24万
  • 依托单位:
  • 依托单位国家:
    美国
  • 项目类别:
    Standard Grant
  • 财政年份:
    2003
  • 资助国家:
    美国
  • 起止时间:
    2003-08-01 至 2007-07-31
  • 项目状态:
    已结题

项目摘要

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.
第一个研究领域是定义在由参考立方体三线性映射得到的不规则六面体单元上的几类有限单元空间的逼近性质。这样的空间被用来近似三维向量函数,并自然出现在许多应用中,包括近似的麦克斯韦方程和使用混合和最小二乘有限元方法的二阶椭圆方程。本文的研究目的是精确地确定最优阶逼近所需的条件,并构造具有这一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有限元逼近。 我们的目的是获得新的局部误差估计,以了解这类有前途的逼近方案的方法以及为扩散主导和对流主导的二阶偏微分方程。研究的第三个领域是使用现在发展良好的理论近似的Reissner-Mindlin板模型(用于研究弯曲的薄板在外部载荷下)作为基础,开发新的方法来使用有限元法近似的弹性壳。 板模型和壳模型在更大程度上都存在“锁定”问题,当应用标准有限元近似方案时,导致薄板和壳的近似性差。 最后一个研究领域涉及为爱因斯坦方程设计有效的数值方法,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某些相同特征的更简单的模型问题来理解爱因斯坦方程的标准数值方法失败的原因,并帮助设计克服这些问题的方法。使用偏微分方程对物理和生物过程进行数学建模已经成为研究许多重要科学问题的标准方法。由于它通常是不可能解决这样的方程精确,可靠和有效的numericalapproximation计划,它可以在计算机上实现的发展,使这成为一个实用的方法,并在科学和工程的许多领域的进展是中央。 本计画研究各种应用问题之数学模型之“有限元素”型近似格式。 这些包括其中存在对流和扩散的气体和流体的流动、用于对受到施加电流的物体中的电场和磁场进行建模的麦克斯韦方程、薄结构的弯曲(例如,aroof)在外部负载下,以及爱因斯坦的方程,用于模拟大规模天文事件(如黑洞碰撞)的引力辐射发射。 这项工作预计将导致新的和改进的数值方法,供科学家和工程师在应用计算。

项目成果

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Richard Falk其他文献

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  • DOI:
    10.1007/bf02694324
  • 发表时间:
    1977-05-01
  • 期刊:
  • 影响因子:
    1.400
  • 作者:
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  • 通讯作者:
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The Kahan Commission Report on the Beirut Massacre
  • DOI:
    10.1007/bf00246008
  • 发表时间:
    1984-04-01
  • 期刊:
  • 影响因子:
    1.100
  • 作者:
    Richard Falk
  • 通讯作者:
    Richard Falk
Ending the Death Dance
The Structure and process of international law : essays in legal philosophy, doctrine and theory
国际法的结构和过程:法哲学、学说和理论论文

Richard Falk的其他文献

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{{ truncateString('Richard Falk', 18)}}的其他基金

Finite Element Approximation of Partial Differential Equations
偏微分方程的有限元逼近
  • 批准号:
    0910540
  • 财政年份:
    2009
  • 资助金额:
    $ 17.24万
  • 项目类别:
    Standard Grant
Finite Element Approximation of Partial Differential Equations
偏微分方程的有限元逼近
  • 批准号:
    0609755
  • 财政年份:
    2006
  • 资助金额:
    $ 17.24万
  • 项目类别:
    Standard Grant
Finite Element Approximation of Problems in Solid Mechanics
固体力学问题的有限元逼近
  • 批准号:
    0072480
  • 财政年份:
    2000
  • 资助金额:
    $ 17.24万
  • 项目类别:
    Standard Grant
Finite Element Methods for Problems in Solid Mechanics
固体力学问题的有限元方法
  • 批准号:
    9704556
  • 财政年份:
    1997
  • 资助金额:
    $ 17.24万
  • 项目类别:
    Standard Grant
Mathematical Sciences: Finite Element Methods for Problems in Solid Mechanics
数学科学:固体力学问题的有限元方法
  • 批准号:
    9403552
  • 财政年份:
    1994
  • 资助金额:
    $ 17.24万
  • 项目类别:
    Continuing Grant
Mathematical Sciences: Finite Element Methods for Partial Differential Equations
数学科学:偏微分方程的有限元方法
  • 批准号:
    9106051
  • 财政年份:
    1991
  • 资助金额:
    $ 17.24万
  • 项目类别:
    Continuing Grant
Mathematical Sciences: Finite Element Methods for Partial Differential Equations
数学科学:偏微分方程的有限元方法
  • 批准号:
    8902120
  • 财政年份:
    1989
  • 资助金额:
    $ 17.24万
  • 项目类别:
    Continuing Grant
Mathematical Sciences: Finite Element Methods for Constrained and Ill-Posed Variational Problems
数学科学:约束和不适定变分问题的有限元方法
  • 批准号:
    8703354
  • 财政年份:
    1987
  • 资助金额:
    $ 17.24万
  • 项目类别:
    Continuing Grant
Mathematical Sciences Research Equipment
数学科学研究设备
  • 批准号:
    8505016
  • 财政年份:
    1985
  • 资助金额:
    $ 17.24万
  • 项目类别:
    Standard Grant
Mathematical Sciences: Finite Element Methods for Constrained and Ill-Posed Variational Problems
数学科学:约束和不适定变分问题的有限元方法
  • 批准号:
    8402616
  • 财政年份:
    1984
  • 资助金额:
    $ 17.24万
  • 项目类别:
    Standard Grant

相似国自然基金

毛竹MLE(mariner-like element)转座酶催化机理研究
  • 批准号:
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  • 批准年份:
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Developments and Applications of Numerical Verification Methods for Finite Element Approximation of Differential Equations
微分方程有限元逼近数值验证方法的发展与应用
  • 批准号:
    23K03232
  • 财政年份:
    2023
  • 资助金额:
    $ 17.24万
  • 项目类别:
    Grant-in-Aid for Scientific Research (C)
Direct and inverse approximation for finite element solutions of the p and h-p versions and applications to three-dimensional propblem, nonlinear problems and Kirchhoff plate problem.
p 和 h-p 版本的有限元解的直接和逆近似以及在三维问题、非线性问题和基尔霍夫板问题中的应用。
  • 批准号:
    RGPIN-2014-04642
  • 财政年份:
    2017
  • 资助金额:
    $ 17.24万
  • 项目类别:
    Discovery Grants Program - Individual
Direct and inverse approximation for finite element solutions of the p and h-p versions and applications to three-dimensional propblem, nonlinear problems and Kirchhoff plate problem.
p 和 h-p 版本的有限元解的直接和逆近似以及在三维问题、非线性问题和基尔霍夫板问题中的应用。
  • 批准号:
    RGPIN-2014-04642
  • 财政年份:
    2016
  • 资助金额:
    $ 17.24万
  • 项目类别:
    Discovery Grants Program - Individual
Direct and inverse approximation for finite element solutions of the p and h-p versions and applications to three-dimensional propblem, nonlinear problems and Kirchhoff plate problem.
p 和 h-p 版本的有限元解的直接和逆近似以及在三维问题、非线性问题和基尔霍夫板问题中的应用。
  • 批准号:
    RGPIN-2014-04642
  • 财政年份:
    2015
  • 资助金额:
    $ 17.24万
  • 项目类别:
    Discovery Grants Program - Individual
Direct and inverse approximation for finite element solutions of the p and h-p versions and applications to three-dimensional propblem, nonlinear problems and Kirchhoff plate problem.
p 和 h-p 版本的有限元解的直接和逆近似以及在三维问题、非线性问题和基尔霍夫板问题中的应用。
  • 批准号:
    RGPIN-2014-04642
  • 财政年份:
    2014
  • 资助金额:
    $ 17.24万
  • 项目类别:
    Discovery Grants Program - Individual
Finite Element Approximation of Partial Differential Equations
偏微分方程的有限元逼近
  • 批准号:
    0910540
  • 财政年份:
    2009
  • 资助金额:
    $ 17.24万
  • 项目类别:
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A posteriori analysis of the finite element approximation for nonlinear parabolic problems
非线性抛物线问题有限元近似的后验分析
  • 批准号:
    18740046
  • 财政年份:
    2006
  • 资助金额:
    $ 17.24万
  • 项目类别:
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Finite Element Approximation of Partial Differential Equations
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NONLINEAR FINITE ELEMENT APPROXIMATION OF FIRST-ORDER PDE'S IN L1
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