  For the finite element solution of Poisson's equation, a local a posteriori
error estimation based on the Hypercircle method is proposed. Even for the
solution of Poisson's equation without the $H^2$ regularity, this method can
provide explicit local error estimation. The efficiency of the proposed method
is demonstrated by numerical experiments for the boundary value problem of
Poisson's equation defined on the 2D and 3D domains.
