In this study, based on the Hypercircle method, an explicit local ${\em a ~ posteriori}$ error estimation for the finite element solution of Poisson's equation is proposed. Compared with classical research, which mainly focuses on the qualitative analysis (convergence rate etc.) of global or local errors, our newly developed method can provide concrete value of error estimation for the approximate solution over subdomain of interest. Even for the solution of Poisson's equation without the $H^2$ regularity, this method can provide explicit local error estimation. Our proposed method is mainly based on the error estimation theorem developed by Liu and Oishi (SIMNA, 2013, 635-722), where the Hypercircle method plays an important role. The Hypercircle method is firstly proposed by Prager and Synge (Quart. Appl. Math., 1947, 1-21) for the purpose of elastic analysis. Fujita proposed a pointwise estimation method for boundary value problems using the $T-T*$ method (J. Phys. Soc. Jpn., 1955,1-8), which is essentially the same as the Hypercircle method. The Hypercircle method has also been applied to the global error estimation; see the work of Braess (Finite elements theory, fast solver, and applications in solid mechanics, 2006) and Neittaanmäki -- Repin (Reliable methods for computer simulation: error control and a posteriori estimates, 2004). In this study, we apply the Hypercircle method to the local error estimation. In this talk, the efficiency of the proposed method will be demonstrated by numerical experiments for the boundary value problem of Poisson's equation defined on the square domain and the L-shaped domain.