If the Solutionof a Given Boundary Value Problem Satisfies a Maximum Principle, Then Aproperly Designed Approximation Should Behave In the Same Way. a Numericalscheme That Does Not Generate Spurious Global Extrema In the Interior of Thecomputational Domain Is Said to Satisfy a Discrete Maximum Principle (Dmp). As In the Continuous Case, Theprecise Formulation of This Criterion Is Problem-Dependent. In Particular, Thezero Row Sum Property (Second Rule from Section 1.6.3) Has the Sameimplications As the Constraint ·V _0 In Continuous Maximum Principles. In the Contextof Finite Difference Approximations to Linear Elliptic Problems, Sufficientconditions of Dmp Were Formulated and Proven By Varga [340] As Early As In1966. These Conditions Are Related to the Concept of Monotone Operators And, In Particular, M-Matrices Which Play an Important Role In Numerical Linearalgebra [339, 354]. a General Approach to Dmp Analysis For Finite Differenceoperators Was Developed By Ciarlet [63]. Its Extension to Finite Elements In[64] Features a Proof of Uniformconvergence, As Well As Simple Geometric Conditions That Ensure Thevalidity of Dmp For a Piecewise-Linear Galerkin Discretization of the (Linear)Model Problem.