This paper considers the finite element solution of the boundary value problem of Poisson's equation and proposes a guaranteed em a posteriori local error estimation based on the hypercircle method. Compared to the existing literature on qualitative error estimation, the proposed error estimation provides an explicit and sharp bound for the approximation error in the subdomain of interest, and its efficiency can be enhanced by further utilizing a non-uniform mesh. Such a result is applicable to problems without $H^2$-regularity, since it only utilizes the first order derivative of the solution. The efficiency of the proposed method is demonstrated by numerical experiments for both convex and non-convex 2D domains with uniform or non-uniform meshes.

翻译：本文件考虑了Poisson方程式边界值问题的有限要素解决方案,并提出了基于高圆圈法的有保证的事后局部误差估计。与关于定性误差估计的现有文献相比,拟议的误差估计为子界域的近似误差提供了明确和尖锐的界限,其效率可以通过进一步使用非统一的网目加以提高。这种结果适用于没有H2美元规则的问题,因为它只使用解决办法的第一级衍生物。对具有统一或非统一的模类的螺旋形和无螺旋2D域进行数字实验,可以证明拟议方法的效率。