Algorithms and Numerical Analysis for Partial Differential Equations

偏微分方程的算法和数值分析

• 批准号: 0310539
• 负责人: Lars Wahlbin
• 金额: --
• 依托单位: Cornell University
• 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
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0310539&HistoricalAwards=false
• 关键词: Algorithms; Numerical; Analysis; Partial; Differential

Wahlbin Lars Wahlbin and Alfred Schatz first continue their study ofasymptotically exact a posteriori error estimators for thepointwise error in approximating the gradient on each simplex bythe finite element method. The problems they consider involvesolutions with singularities that pose new difficulties. Whatmakes these problems difficult is that one wants totally localinformation about errors when, in fact, the solution and itsapproximation are globally influenced in particular by thesingularities. The investigators also look into a priori errorestimates for second order elliptic partial differentialequations with discontinuous coefficients on nonconvex polygonaldomains, allowing meshes that are highly refined. The estimatesunder consideration justify the use of refined grids in a varietyof highly singular problems. Because most refined grids areconstructed using self-adaptive codes that may have differentmethods of choosing a grid, no a priori assumption is made aboutthe distribution of the mesh. The work described above is carriedout under the assumption that meshes are shape-regular, butbasically nothing else is assumed concerning the meshes. If somefurther local structure is placed on the mesh (so-called one plusalpha regular meshes), then a limited amount of superconvergencemay occur (as shown by Xu and Zhang in a particular situation).The investigators, together with Zhang, apply this idea ingreater generality, namely to meshes that are perturbations ofthose symmetric with respect to a point. The symmetry theorypredicts superconvergent points in many practical situations andit is important to determine how this stands up underperturbations. If even more structure is given to the mesh,namely, that it is translation invariant, difference quotientsmay be used for approximating derivatives. The investigatorsconstruct new compact forms of such difference quotients.Finally, the investigators consider finite element methods forvisco-elastic problems. The investigators look into basic properties of the finiteelement approximation method, widely used in engineering andscience practice, in industry, in research laboratories as wellas in academia. The finite element method builds an approximationto the solution of a differential equation on a region bybreaking the region into smaller pieces and approximating thesolution on each separate piece, say as a combination offunctions that is easy to compute. Pasting together the separateapproximations gives a global approximation to the solution. Themethod has its roots in the airplane industry and started to beused in all branches of science and engineering in tne 1960's.Well over a hundred thousand articles have been written on thismethod. The investigators' aim in pursuing the fundamentalbehavior of this method is two-fold: to get a deeperunderstanding for what the method does, as currently practiced,and to use this understanding to develop new and better methods.A particular aspect that the investigators consider is that ofreliably gauging how well the computer method approximates theunderlying engineering or scientific model, whose exact solutionis of course unknown.

瓦尔宾 Lars Wahlbin和Alfred Schatz首先继续他们的研究渐近精确的后验误差估计的逐点误差近似梯度的每个单形的有限元方法。 他们考虑的问题涉及到奇异性的解决方案，这带来了新的困难。 使这些问题变得困难的是，人们希望完全本地的信息错误时，事实上，解决方案和它的近似是全球特别是奇异性的影响。 研究人员还研究了非凸域上具有不连续系数的二阶椭圆型偏微分方程的先验误差估计，允许高度精细的网格。 在考虑下的估计证明了在各种高度奇异的问题中使用精化网格的合理性。 由于大多数精细网格是使用自适应代码构造的，这些代码可能具有不同的网格选择方法，因此没有对网格的分布进行先验假设。 上述工作是在网格形状规则的假设下进行的，但基本上没有对网格作任何其他假设。 如果在网格上再加上一些局部结构（所谓的一加α正则网格），那么就可能出现有限的超收敛（如徐和张在特定情况下所示）。 对称性理论预测超收敛点在许多实际情况下，它是重要的，以确定如何站在扰动下。 如果给网格更多的结构，即它是平移不变的，则可以使用差分导数来近似导数。 最后，研究者考虑了粘弹性问题的有限元方法。 研究人员研究了有限元逼近方法的基本特性，该方法广泛应用于工程和科学实践、工业、研究实验室以及学术界。 有限元法通过将区域分成更小的块并在每个单独的块上近似解来建立微分方程在区域上的近似解，例如作为易于计算的函数的组合。 把各个近似值粘在一起就得到了解的全局近似值。 这种方法起源于飞机制造业，并于20世纪60年代开始在科学和工程的所有分支中使用。 研究者追求这种方法的基本行为的目的是双重的：更深入地理解这种方法的作用，就像目前实践的那样，并利用这种理解来开发新的和更好的方法。研究者考虑的一个特别方面是可靠地衡量计算机方法对基本工程或科学模型的近似程度，当然，其确切的解决方案是未知的。