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

这项研究的主要目的是设计，分析和测试有效的，可靠的，和强大的后验误差估计的各种有限元近似计算具有挑战性的问题，也就是说，这些问题具有以下现象：界面奇异性，不连续性（在形式的冲击前，和内部和边界层），和/或振荡的各种尺度（多尺度现象）。在这个项目中开发的估计器不使用这些现象的位置和特性的先验知识，它们可以更容易地应用于高度非线性问题，并有可能被应用到复杂的系统中产生的应用。在这个项目中要调查的估计是恢复型的恢复估计拥有一些有吸引力的功能，导致他们在工程实践中广泛采用，并以数学研究的主题。然而，现有的恢复估计有几个主要缺点更具挑战性的问题。研究人员和他的同事们通过引入创新的恢复程序和开发一种方法来克服这些障碍，该方法涉及如何为复杂系统设计高效，可靠和强大的恢复估计器。该方法被应用到各种问题所产生的连续介质力学设计鲁棒估计，将在理论和numbers.Self-adaptive数值方法研究提供了一个强大的和自动化的方法在科学计算。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，已成为复杂自然和工程问题计算机模拟的必要工具。正如美国国家研究理事会所指出的那样，AMR是解决计算上具有挑战性的问题的两个必要工具之一（AMR和并行计算机）。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的全局和局部误差源。该项目的成功将增强AMR算法自动定位物理界面、检测层和不连续性以及解决各种尺度振荡的能力。在该项目中开发的方法将阐明如何设计不确定的问题，如对流为主的扩散问题相对粗糙的网格估计。对这些不确定性问题的研究是完全开放的，不仅在技术上，而且在概念上都需要重大突破。

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.