Efficient, Reliable, and Robust A Posteriori Error Estimators of Recovery Type

本研究的主要目的是设计、分析和测试高效、可靠和稳健的后验误差估计器，用于各种具有计算挑战性的有限元近似；也就是说，那些具有以下现象的问题：界面奇点，不连续（以激波锋面，内层和边界层的形式），和/或各种尺度的振荡（多尺度现象）。在本项目中开发的估算器不使用这些现象的位置和特征的先验知识；然后，它们可以更容易地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。本项目拟调查的估算机构属回收类；采收率估计器具有许多吸引人的特点，这些特点使其在工程实践中得到广泛采用，并成为数学研究的主题。然而，对于更具挑战性的问题，现有的采收率估算器存在几个主要缺陷。研究者和他的同事们克服了这些障碍，引入了一种创新的恢复程序，并开发了一种方法，用于设计高效、可靠和健壮的复杂系统的恢复估计器。该方法应用于连续介质力学中出现的各种问题，以设计将在理论上和数值上进行研究的稳健估计器。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，已成为计算机模拟复杂自然和工程问题的必要工具。正如美国国家研究委员会所确定的那样，AMR是两个必要的工具之一（AMR和并行计算机），用于计算重大的挑战性问题。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。这个项目的成功将使AMR算法能够自动定位物理界面，检测层和不连续，以及解决各种尺度的振荡。该项目中开发的方法将阐明如何为不确定问题设计估计器，例如相对粗糙网格上的对流主导扩散问题。对这些不确定问题的研究是完全开放的，不仅需要在技术上而且需要在概念上取得重大突破。

The main aim of this proposed research is to design, analyze, and test efficient, reliable, and robust a posteriori error estimators for various finite element approximations to computationally challenging problems; that is, those problems having the following phenomena: interface singularities, discontinuities (in the form of shock-like fronts, and of interior and boundary layers), and/or oscillations of various scales (multiscale phenomena). Estimators to be developed in this project do not use a priori knowledge of locations and characteristics of these phenomena; they may then be applied more readily to highly nonlinear problems and have potential to be applied to complex systems arising in applications. Estimators to be investigated in this project are of the recovery type; recovery estimators possess a number of attractive features that have led to their widespread adoption in engineering practice and to the subject of mathematical study. However, existing recovery estimators have several major drawbacks for more challenging problems. The investigator and his colleagues overcome those obstacles by introducing an innovative recovery procedure and developing a methodology on how to design efficient, reliable, and robust recovery estimators for complex systems. The methodology is applied to various problems arising from continuum mechanics to design robust estimators that will be studied theoretically and numerically.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 necessary tool in computer simulations of complex natural and engineering problems. As identified by the US National Research Council, AMR is one of two necessary tools (AMR and Parallel Computer) for computationally grand challenging problems. The key ingredient for success of AMR algorithms are a posteriori error estimates that are able to accurately locate sources of global and local error in the current approximation. Success in this project will empower the ability of AMR algorithms for automatically locating physical interfaces, detecting layers and discontinuities, and resolving oscillations of various scales. The methodology developed in the project will shed light on how to design estimators for indefinite problems such as convection-dominant diffusion problems on relatively coarse meshes. Research on those indefinite problems is completely open and requires major breakthrough not only technically but also conceptually.