Many practical problems occur due to the boundary value problem. This paper evaluates the finite element solution of the boundary value problem of Poissons equation and proposes a novel 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 is applicable to problems without the $H^2$ regularity. The efficiency of the proposed method is demonstrated by numerical experiments for both convex and non-convex 2D domains.
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