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。这种估计在实践中不仅用于可靠地确定近似解何时足够精确，而且还用于（有效地）自适应地改进近似。该方法考虑了辅助子空间误差估计，该估计是通过在辅助空间中计算近似误差函数得到的。这种近似误差函数提供了很大的灵活性，原则上，它可以用于自适应有限，我们在这里关注的估计和自适应性：各种范数的误差，各种范数的高阶导数，一般线性泛函的泛函误差，以及特征值和不变子空间计算中的误差。对于二维低阶有限元和二阶线性椭圆型边值问题，能量范数误差层次估计的鲁棒性已在理论和实践中得到证实。本提案旨在显著扩展理论和实践，不仅包括上述各种误差度量，还包括二维和三维的高阶元素（p-和hp-自适应），以及不同类型的算子和有限元，包括偏微分方程系统。此外，在可能的情况下，还将发展适应性收敛理论。所提出的研究的一个关键组成部分是开发一个基本框架，在这个框架中，通过考虑潜在问题的一些基本性质和用于近似解的空间，为计算近似误差函数的辅助空间的适当选择提供明确的指导。在复合材料建模过程中，自动检测和适应相关精细和粗尺度特征的能力往往是设计这种材料以及在存在复杂介质的情况下进行遥感（例如对自然资源的非破坏性勘探）所必不可少的辅助工具。这一建议涉及到一种通用的和非常灵活的方法的发展，以误差估计和自适应改进的近似值在各种情况下，提供仔细发展的具体情况，如上述提到的。一个清晰的误差估计和自适应的理论框架，加上几个重要的实践实现，不仅有助于从业者在他们的特定环境中做出适当的选择，而且还可以更容易地训练学生能够开发这些工具来解决很少（如果有的话）可用的问题。提案中的各种项目包括国内和国际合作，以及研究生的教育和参与。此外，与此提案相关的大部分软件将由提案人从他的网站上免费提供。

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.