This paper addresses the existence, uniqueness, and approximation of saddle-point problems arising from the method of Lagrangian multipliers. The authors provide necessary and sufficient conditions for the existence and uniqueness of solutions to such problems, which are often encountered in elasticity and other applications. They introduce the concept of "discrete problems" and derive upper bounds for the error between the exact and discrete solutions. The paper also discusses numerical integration and the approximation of non-conforming elements, providing conditions for convergence and error bounds. The theoretical results are illustrated with examples from plate bending problems and hybrid methods. The authors conclude with a discussion on the practical implementation and numerical integration techniques, emphasizing the importance of these methods in solving complex engineering problems.This paper addresses the existence, uniqueness, and approximation of saddle-point problems arising from the method of Lagrangian multipliers. The authors provide necessary and sufficient conditions for the existence and uniqueness of solutions to such problems, which are often encountered in elasticity and other applications. They introduce the concept of "discrete problems" and derive upper bounds for the error between the exact and discrete solutions. The paper also discusses numerical integration and the approximation of non-conforming elements, providing conditions for convergence and error bounds. The theoretical results are illustrated with examples from plate bending problems and hybrid methods. The authors conclude with a discussion on the practical implementation and numerical integration techniques, emphasizing the importance of these methods in solving complex engineering problems.