Flux Recovery, A Posteriori Error Estimation, and Adaptive Finite Element Method

通量恢复、后验误差估计和自适应有限元方法

• 批准号: 0810855
• 负责人: Zhiqiang Cai
• 金额: $ 27.1万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-08-15 至 2012-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0810855&HistoricalAwards=false
• 关键词: Flux; Recovery; Posteriori; Error; Estimation

The main purpose of this project is to develop, analyze, and test novel,accurate a posteriori error estimators of the recovery type for variousfinite element discretizations of a variety of elliptic equations andsystems arising from solid and fluid mechanics, including nonlinearproblems. The investigator and his colleagues plan to study two types ofrecovery procedures: one is accurate only for the constitutive equationand the other is accurate for both the constitutive and equilibriumequations. Based on these recovered fluxes (or the stresses for solidand fluid mechanics), they will study three kinds of estimators. Inparticular, they will study an exact estimator on any given mesh,including an arbitrary initial mesh, with no regularity assumptions.Exactness on any given mesh implies that the estimator is ideallyperfect for error control (or the so-called solution verification) oncoarse (pre-asymptotic) meshes. No regularity assumptions in thisproject mean that the only assumptions on the existence of theunderlying problem are required.This is weaker than those required for approximation theory and muchweaker than those required by the current theory of the recovery-basedestimators. Therefore, the estimators can be applied to problems ofpractical interests such as interface singularities, discontinuities inthe form of shock-like fronts and of interior or boundary layers. Thesecond part of the project is to establish convergence of adaptivefinite element methods based on the recovery-based estimators and thenewly developed estimators of this project.A major problem with computer simulations of physical phenomena is thatall computational results obtained involve numerical error.Discretization error can be large, pervasive, unpredictable by classicalheuristic means, and can invalidate numerical predictions.A posteriori error estimation is a rigorous mathematical theory forestimating and quantifying discretization error in terms of the error'smagnitude and distribution based on the current simulation and givendata of the underlying problem. This information provides bases forsolution verification and for adaptive control of simulation process:adaptive mesh refinement, adaptive control of mathematical models andnumerical algorithms. Success in this project will provide accurate andreliable a posteriori error estimators for a large class of ellipticequations/systems arising from engineering, physics, aerodynamics,atmospheric sciences, geology, biomechanics, material sciences,nano-technology, and industrial applications. The development of theexact estimator will enable error control on pre-asymptotic meshes andpredictable computation analysis. Error control on pre-asymptotic meshesis of paramount importance for simulating physical phenomena inengineering applications and scientific predictions with limitedcomputer resources.

这个项目的主要目的是开发、分析和测试新的、准确的恢复型后验误差估计器，用于固体和流体力学中产生的各种椭圆型方程和系统的各种有限元离散，包括非线性问题。这位研究人员和他的同事们计划研究两种类型的恢复程序：一种只对本构方程准确，另一种对本构方程和平衡方程都准确。基于这些恢复的通量(或固体和流体力学的应力)，他们将研究三种估计器。特别是，他们将研究任何给定网格上的精确估计器，包括任意初始网格，而不需要正则性假设。在任何给定网格上的精确度意味着该估计器对于粗(预渐近)网格上的误差控制(或所谓的解验证)是理想的完美的。这个项目中没有正则性假设，这意味着只需要关于潜在问题存在的假设。这比近似理论所需的假设要弱得多，也比当前基于恢复的估计器的理论所要求的弱得多。因此，这些估计器可以应用于实际问题，如界面奇异性、激波阵面形式的不连续性以及内部或边界层问题。该项目的第二部分是建立基于恢复的估计器和新开发的估计器的自适应有限元方法的收敛。物理现象的计算机模拟的一个主要问题是所有的计算结果都涉及数值误差。离散化误差可能很大，普遍存在，通过经典的启发式方法不可预测，并且可能使数值预测无效。后验误差估计是一种严格的数学理论，根据当前的模拟和给定的潜在问题的数据，根据误差的小范围和分布来预测和量化离散化误差。这些信息为解验证和模拟过程的自适应控制提供了基础：自适应网格加密、数学模型的自适应控制和数值算法。该项目的成功将为工程、物理、空气动力学、大气科学、地质学、生物力学、材料科学、纳米技术和工业应用中的一大类椭圆方程/系统提供准确和可靠的后验误差估计器。精确估计器的发展将使预渐近网格上的误差控制和可预测的计算分析成为可能。预渐近网格的误差控制对于用有限的计算机资源模拟工程应用中的物理现象和科学预测是至关重要的。