  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.
