A Posteriori Error Estimation through Duality and Some Other Topics

自适应数值方法为科学计算提供了一种强大的自动化方法。特别是自适应网格细化（AMR）算法在计算科学和工程中得到了广泛的应用，已成为复杂自然科学和工程问题计算机模拟的常用工具。正如美国国家研究委员会所确定的那样，AMR是在计算上解决大挑战问题的两个必要工具之一（AMR和并行计算）。AMR算法成功的关键因素是后验误差估计，它能够准确地定位当前近似中全局和局部误差的来源。计算机模拟复杂系统的另一个挑战是计算机预测的可靠性。这些考虑（AMR算法的效率和误差控制）表明需要一个误差估计器，它可以从计算的数值解和潜在问题的给定数据中后验提取。理想情况下，这种后验误差估计应该为根据误差大小和空间分布估计和量化离散误差提供一个潜在的严格数学理论。这个项目的成功将允许AMR算法自动定位物理界面，检测层和不连续，并解决各种尺度的振荡。在本项目中开发的对偶估计器将解决一类问题中最自然但又极其困难的离散化误差控制问题，从而部分保证计算机模拟的可靠性。本研究项目的重点是通过对偶方法开发、分析和测试后验误差估计器。本课题所开发的对偶估计具有可靠度保证界，可靠度常数为1。因此，这些估计量对于离散化误差控制是完美的，并且可以作为迭代解算器的精确停止准则。对偶方法可以应用于连续介质力学中产生的大量问题，包括线性和非线性问题。由于这些估计器不会使用关于界面奇点、不连续（以类激波锋面、内层和边界层的形式）和/或各种尺度（多尺度现象）的振荡的位置和特征的先验知识，因此它们可能更容易应用于高度非线性问题，并有可能应用于应用中出现的复杂系统。本文研究的重点和难点在于：(1)明确或局部地构造对偶变量的近似，从而使所得到的指标是高效和鲁棒的；(2)对效率和鲁棒性的理论和数值证实。最后，提出的研究的一小部分解决了一个开放的理论问题，即界面问题估计器的鲁棒性。

Self-adaptive numerical methods provide a powerful and automatic approach in scientific computing. In particular, Adaptive Mesh Refinement (AMR) algorithms have been widely used in computational science and engineering and have become a common tool in computer simulations of complex natural science and engineering problems. As identified by the US National Research Council, AMR is one of two necessary tools (AMR and Parallel Computing) for computationally tackling Grand Challenge problems. The key ingredient for success of AMR algorithms is a posteriori error estimates that are able to accurately locate sources of global and local error in the current approximation. Another challenge in computer simulations of complex systems is the reliability of computer predictions. These considerations (efficiency in AMR algorithms and error control) demonstrate the need for an error estimator that can a posteriori be extracted from the computed numerical solution and the given data of the underlying problem. Such an a posteriori error estimate ideally should provide an underlying rigorous mathematical theory for estimating and quantifying discretization error in terms of the error's magnitude and its spatial distribution. Success in this project will allow AMR algorithms to automatically locate physical interfaces, detect layers and discontinuities, and resolve oscillations of various scales. The dual estimators to be developed in this project will resolve the most natural but extremely difficult question of discretization error control on coarse meshes for a class of problems and hence partially guarantee reliability of computer simulations.This research project focuses on the development, analysis, and test of a posteriori error estimators through the methodology of duality. The dual estimators to be developed in this project will have a guaranteed reliability bound with reliability constant being one. Hence, these estimators are perfect for discretization error control and may be used as an accurate stopping criterion for iterative solvers. The methodology of duality may be applied to a large class of problems arising from continuum mechanics including linear and nonlinear problems. Since these estimators will not use a priori knowledge on the locations and characteristics of interface singularities, discontinuities (in the form of shock-like fronts, and of interior and boundary layers), and/or oscillations of various scales (multi-scale phenomena), they may then be applied more readily to highly nonlinear problems and have the potential of being applied to complex systems arising in applications. The emphases and the difficulties of the proposed research are (1) explicit or local construction of an approximation to the dual variable such that the resulting indicator is efficient and robust, and (2) theoretical and numerical confirmation of the efficiency and robustness. Finally, a small portion of the proposed research addresses an open theoretical question on the robustness of estimators for interface problems.