Averaging techniques for the effective numerical solution of Symm's integral equation of the first kind
Averaging techniques for the effective numerical solution of Symm's integral equation of the first kind
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DOI：
10.1137/040609033
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发表时间：
2006-01-01
影响因子：
3.1
通讯作者：
Praetorius, D
中科院分区：
文献类型：
--
作者：
Carstensen, C;Praetorius, D
用于有限元误差控制的平均技术，有时被称为用于梯度恢复的ZZ估计器，由于其惊人的简单性和普适性而在工程上享有很高的受欢迎程度：人们甚至不需要PDE来应用不昂贵的后处理例程。最近，平均技术已被数学证明是可靠和有效的各种应用的有限元方法。本文建立了一类边界积分法的平均误差估计。以含非局部单层积分算子的第一类Symm积分方程作模型方程，从理论和数值两方面进行了研究。我们引入了四个新的误差估计器，它们被证明是可靠和有效的，直到高阶。高阶项取决于精确解的正则性。几个数值实验验证了理论结果，并表明所提出的估计器可以很好地估计[正常未知]误差，即误差和估计器几乎重合。
Averaging techniques for finite element error control, occasionally called ZZ estimators for the gradient recovery, enjoy a high popularity in engineering because of their striking simplicity and universality: One does not even require a PDE to apply the nonexpensive post-processing routines. Recently, averaging techniques have been mathematically proved to be reliable and efficient for various applications of the finite element method. This paper establishes a class of averaging error estimators for boundary integral methods. Symm's integral equation of the first kind with a nonlocal single-layer integral operator serves as a model equation studied both theoretically and numerically. We introduce four new error estimators which are proven to be reliable and efficient up to terms of higher order. The higher-order terms depend on the regularity of the exact solution. Several numerical experiments illustrate the theoretical results and show that the [ normally unknown] error is sharply estimated by the proposed estimators, i.e., error and estimators almost coincide.