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

本项目的主要目的是开发、分析和测试用于固体和流体力学中各种椭圆方程和系统的各种有限元离散化的新颖、准确的后验误差估计器，包括非线性问题。研究者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的精确估计，包括任意初始网格，没有规则假设。任何给定网格上的准确性意味着估计器对于粗糙（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。在这个项目中，没有规则假设意味着只需要对潜在问题的存在进行假设。这比近似理论所要求的要弱，比当前基于恢复的估计理论所要求的要弱得多。因此，这些估计量可以应用于实际问题，如界面奇点、类激波锋面和内层或边界层形式的不连续。项目的第二部分是在基于恢复的估计量和本项目新开发的估计量的基础上建立自适应有限元方法的收敛性。用计算机模拟物理现象的一个主要问题是所有得到的计算结果都包含数值误差。离散化误差可以是大的，普遍的，不可预测的经典启发式方法，并可以使数值预测无效。后验误差估计是一种严格的数学理论，它根据当前的模拟和给定的潜在问题的数据，根据误差的大小和分布对离散误差进行预测和量化。这些信息为解验证和仿真过程的自适应控制提供了基础：自适应网格细化、数学模型和数值算法的自适应控制。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等领域的大量椭圆方程/系统提供准确可靠的后验误差估计。精确估计器的开发将实现对预渐近网格的误差控制和可预测的计算分析。预渐近网格的误差控制对于模拟工程应用中的物理现象和有限计算机资源下的科学预测具有至关重要的意义。

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.