Algorithms and Numerical Analysis for Partial Differential Equations

Wahlbin、Lars Wahlbin和Alfred Schatz首先研究了用有限元方法逼近每个单纯形上的梯度时逐点误差的渐近精确的后验误差估计。他们考虑的问题涉及具有奇点的解，这带来了新的困难。使这些问题变得困难的是，人们想要关于错误的全部局部信息，而事实上，解及其近似尤其受到这些奇性的全局影响。研究人员还研究了非凸多边形域上具有不连续系数的二阶椭圆型偏微分方程解的先验误差估计，从而允许高度加密的网格。考虑中的估计证明了在各种高度奇异的问题中使用精化网格是合理的。由于大多数精化网格是使用自适应代码构建的，而自适应代码可能有不同的网格选择方法，因此没有对网格的分布做出先验假设。上述工作是在网格是形状规则的假设下进行的，但基本上没有假设其他关于网格的情况。如果在网格上放置更多的局部结构(所谓的一个正则网格)，则可能会出现有限数量的超收敛(如徐和张在特定情况下所示)。研究人员与张一起将这一思想应用于更一般的情况，即关于点对称的网格的扰动。对称性理论在许多实际情况下预测了超收敛点，这对于确定它在扰动下如何站得住脚是很重要的。如果给网格赋予更多的结构，即它是平移不变的，则差商可以用来逼近导数。研究人员构造了这种差商的新的紧致形式。最后，研究人员考虑了粘弹性问题的有限元方法。研究人员研究了有限元近似方法的基本性质，这种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。有限元方法通过将区域分解成较小的块，并在每个单独的块上近似解，例如作为易于计算的函数的组合，来建立对区域上的微分方程解的近似。将分离的近似值粘贴在一起，可以得到解的全局近似值。这种方法起源于飞机工业，并在20世纪60年代开始在科学和工程的各个分支中使用。关于这种方法的文章已有10万多篇。研究人员追求这种方法的基本行为有两个目的：更深入地了解该方法的作用，就像目前所实践的那样，并利用这种理解来开发新的和更好的方法。研究人员考虑的一个特定方面是可靠地衡量计算机方法与潜在的工程或科学模型的接近程度，其确切的解决方案当然是未知的。

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.