On hypercircle-based a posteriori error estimation of finite element solutions

This paper first reviews the hypercircle method for a posteriori error estimation of finite element (FE) solutions to Poisson’s equation, which originates from the classical Prager–Synge theorem and employs basic P1 and H(div) elements. With new numerical results, we examine a simplified post-processing method for P1 solutions that avoids mixed FE systems. While this approach offers practical advantages, it lacks rigorous error analysis. Although the numerical results are promising, the theoretical foundation remains open, and further investigation is needed.