Recovery Type A Posteriori Error Estimates

张 研究了偏微分方程数值解的后验误差估计. 他发展了恢复型后验误差估计的一般数学理论，并探索了一种新的梯度恢复技术和相关的误差估计，具有比现有的更好的性能。 本文采用了有限元超收敛理论、渐近正则性理论以及改进的内点分析理论。 微分方程的数值解在科学和工程的各个领域都是必不可少的。 但数值解往往不如人们所希望的那样精确。 因此，重要的是估计误差，并使用该估计，以提高精度。 误差估计是有效可靠计算的基础。 恢复型后验误差估计是一种后处理技术，它利用由某种基方法产生的数值解来产生比基方法解精度更高的近似解，并将这两种解联合收割机结合起来估计近似解中的误差。 这些技术是常见的，但在理论上还没有得到理解或证明，特别是对于非均匀网格。 研究者对这些方法进行了研究，并提出了一种新的恢复型误差估计器，它比现有的方法具有一些优点。 该项目填补了工程实践与数学理论发展之间的空白。 该项目包括研究生的培训，与工程师和其他科学计算学科以及工业界有跨学科的联系。

Zhang The investigator studies a posteriori error estimates forthe numerical solution of partial differential equations. Hedevelops general mathematical theory for recovery type aposteriori error estimates and explores a new gradient recoverytechnique and associated error estimator that has betterproperties than the existing ones. Some recent mathematicaltheory in finite element superconvergence, asymptotic regularity,as well as improved interior analysis, is employed in theproject. Numerical solutions of differential equations are essentialin all areas of science and engineering. But the numericalsolutions are often less accurate than one desires. It istherefore important to estimate the error and to use thatestimate to improve the accuracy. Error estimation is essentialfor effective and reliable computation. Recovery type aposteriori error estimates are post-processing techniques thatuse a numerical solution produced by some base method to produceapproximations that are of higher order of accuracy than the basemethod solution, and that combine the two solutions to estimatethe error in the approximate solution. The techniques are commonbut not theoretically understood or justified, especially fornonuniform meshes. The investigator studies these methods anddevelops a new kind of recovery type error estimator that offerssome advantages over present alternatives. The project fills agap between engineering practice and mathematical theoreticaldevelopment. The project includes training of graduate students.There are interdisciplinary connections with engineers and otherscientific computing disciplines, and with industry.