In this paper we give the necessary and sufficient conditions for the Discrete Maximum Principle (DMP) to hold. We prove the convergence of the nonlinear finite element method applied to the logistic equation by using that the Jacobian matrix evaluated in the supersolution, provided by the a priori bound, is a non-singular M-matrix, which is proved in a fast way using both, the positiveness of its principal eigenvalue and the DMP. Meanwhile a positive subsolution provides the coercivity constant. The numerical simulations show that the nonlinear finite element approximate solutions do not oscillate if the DMP is fulfilled. The characterization of the DMP and the mesh sizes guaranteeing the existence of positive sub- and supersolutions of the nonlinear finite element approximate problem, in the case of variable coefficients and all types of boundary conditions are some of the novelties of this paper. The excellent performance of the method is tested in two examples with boundary layers caused by very small diffusion.

在本文中，我们给出了满足离散最大原理（DMP）的充要条件。通过使用先验界线提供的在超级解决方案中评估的雅可比矩阵是一个非奇异的M矩阵，我们证明了应用于逻辑方程的非线性有限元方法的收敛性。其主要特征值和DMP的正性。同时，正子解提供了矫顽力常数。数值模拟表明，如果满足DMP要求，非线性有限元近似解就不会振动。DMP的特征和网格尺寸可确保存在非线性有限元近似问题的正子解和超级解，在变系数和所有类型的边界条件的情况下，本文都有一些新颖之处。在两个示例中测试了该方法的出色性能，这些示例的边界层由很小的扩散引起。