In order to increase the accuracy and the stability of a scheme dedicated to the approximation of diffusion operators on any type of grids, we propose a method which locally reduces the curvature of the discrete solution where the loss of monotony is observed. The discrete solution is shown to fulfill a variational formulation, thanks to the use of Lagrange multipliers. We can then show its convergence to the solution of the continuous problem, and an error estimate is derived. A numerical method, based on Uzawa's algorithm, is shown to provide accurate and stable approximate solutions to various problems. Numerical results show the increase of precision due to the application of the method.