Investigation of Auxiliary Subspace Techniques as a General Tool for A Posteriori Error Estimation

后验误差估计是高性能有限元计算的重要组成部分。这样的估计在实践中不仅用于可靠地确定近似解何时足够准确，而且还用于(有效地)自适应地改进近似。该方案考虑了辅助子空间误差估计，这些误差估计是通过计算辅助空间中的近似误差函数而得到的。这种近似误差函数在原则上为如何将其用于自适应有限元提供了很大的灵活性，我们在这里关注关于以下方面的估计和适应性：各种范数中的误差、各种范数中的高阶导数、一般类线性泛函的函数误差、以及特征值和不变子空间计算中的误差。对于低阶有限元和二维线性椭圆型边值问题，能量范数误差的分级误差估计的稳健性在理论和实践中都是公认的。这一建议旨在显著扩展理论和实践，不仅包括上述各种误差度量，而且还包括二维和三维(p-和hp-自适应)的高阶元素，以及不同类型的算子和有限元，包括偏微分方程组。此外，在可能的情况下，还将开发一种自适应收敛理论。拟议研究的一个关键组成部分是制定一个基本框架，其中通过考虑基本问题的一些基本性质和用于近似解的空间，就计算近似误差函数的适当辅助空间的适当选择提供明确的指导。自动检测和适应复合材料建模中的相关精细和粗略特征的能力往往是协助此类材料设计以及在存在复杂介质(如自然资源的非破坏性勘探)的情况下进行遥感的必不可少的能力。这项建议涉及在各种情况下制定一种一般性的、非常灵活的方法来估计误差和对近似值进行适应性改进，从而仔细制定感兴趣的具体案例，如上文提到的那些案例。一个清晰的错误估计和适应性的理论框架，以及几个重要的实践实现，不仅将帮助实践者在其特定的环境中做出适当的选择，而且将使学生更容易训练他们能够开发这种工具来解决几乎没有(如果有的话)的问题。提案中的各种项目包括国家和国际合作，以及研究生的教育和参与。此外，与本提案一起开发的许多软件将由提案人在其网站上免费提供。

A posteriori error estimation is an essential component of high-performance finite element computations. Such estimates are used in practice not only to reliably determine when an approximate solution is accurate enough, but also to (efficiently) adaptively improve the approximation. This proposal considers auxiliary subspace error estimates, which are derived from computing an approximate error function in an auxiliary space. Such an approximate error function provides great flexibility, in principle, in how it may be used for adaptive finite, and we are here concerned estimation of and adaptivity with respect to: error in a variety of norms, higher-order derivatives in a variety norms, functional error for general classes of linear functionals, and error in eigenvalue and invariant subspace computations. The robustness of hierarchical error estimates of energy-norm error is well-established both in theory and in practice for low-order finite elements and second-order linear elliptic boundary value problems in two dimensions. This proposal aims to significantly extend both theory and practice not only to include the various error measures mentioned above, but also higher-order elements in two and three dimensions (p- and hp-adaptivity), as well as to different types of operators and finite elements, including systems of partial differential equations. Additionally, an adaptive convergence theory would also be developed, where possible. A key component of the proposed research is the development of a basic framework in which clear guidance concerning an appropriate choice of auxiliary space for computing the approximate error function is provided by considering a few basic properties of the underlying problem and the space which was used for the approximate solution.The ability to automatically detect and adapt to relevant fine and coarse-scale features in the modeling of composite materials is often indispensable as an aid for design of such materials, as well as for remote sensing in the presence of complex media (e.g. non-destructive exploration for natural resources). This proposal concerns the development of a general and very flexible approach to error estimation and adaptive improvement of approximations in a variety of contexts, providing careful development of specific cases of interest, such as those mentioned above. A clear theoretical framework for error estimation and adaptivity, together with several important practical realizations, will not only aid practitioners in making appropriate choices in their particular contexts, but will also make it easier to train students to be able to develop such tools for problem where little (if any) are available. The various projects in the proposal include both national and international collaboration,as well as the education and involvement of graduate students. Additionally, much of the software developed in conjunction with this proposal will be made freely available by the proposer from his website.