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.

This study aims to derive an integrable discretization of the Bernoulli equation that achieves arbitrary higher-order accuracy while preserving the original solution structure. Using the Padé approximation, we develop discretizations for nonhomogeneous first-order linear differential equations with both constant and variable coefficients. By transforming the independent variable, we obtain the discretization of the Bernoulli equation and its general solution.