This paper introduces a mixed finite element method for solving second-order elliptic problems. The authors, P.A. Raviart and J.M. Thomas, focus on a variational formulation known as the complementary energy principle, which involves finding a discrete function \( p_h \in W_h \) that minimizes the complementary energy functional \( I(q) \) over an affine manifold \( W \) of vector-valued functions satisfying the equilibrium equation \( \text{div } q + f = 0 \). The practical construction of the submanifold \( W_h \) is challenging due to the need for explicit solutions of the equilibrium equation in the entire domain \( \Omega \). To address this issue, the authors propose a more general variational principle, the Hellinger-Reissner principle, which removes the equilibrium equation by introducing a Lagrange multiplier. This mixed method is particularly useful in practical problems and has been applied to nonlinear problems of radiative transfer. The paper outlines the derivation of the mixed variational formulation, the definition of related discrete elements, and the error analysis of the finite element method. It also generalizes the results to mixed methods using rectangular elements. The paper references several key works on mixed methods and provides detailed notations for Sobolev spaces and their norms.This paper introduces a mixed finite element method for solving second-order elliptic problems. The authors, P.A. Raviart and J.M. Thomas, focus on a variational formulation known as the complementary energy principle, which involves finding a discrete function \( p_h \in W_h \) that minimizes the complementary energy functional \( I(q) \) over an affine manifold \( W \) of vector-valued functions satisfying the equilibrium equation \( \text{div } q + f = 0 \). The practical construction of the submanifold \( W_h \) is challenging due to the need for explicit solutions of the equilibrium equation in the entire domain \( \Omega \). To address this issue, the authors propose a more general variational principle, the Hellinger-Reissner principle, which removes the equilibrium equation by introducing a Lagrange multiplier. This mixed method is particularly useful in practical problems and has been applied to nonlinear problems of radiative transfer. The paper outlines the derivation of the mixed variational formulation, the definition of related discrete elements, and the error analysis of the finite element method. It also generalizes the results to mixed methods using rectangular elements. The paper references several key works on mixed methods and provides detailed notations for Sobolev spaces and their norms.