Discrete maximum principles for the Courant finite element solution of some nonlinear elliptic problems

Abstract. The discrete maximum principle (DMP) is an important measure of the qualitative reliability of the numerical solutions for elliptic PDE models. We are motivated by known results on the DMP and nonnegativity preservation of finite element (FE) solutions under the condition that sufficiently small enough mesh size \(h\) is used. We extend the above results by explicitly looking for how much the mesh size \(h\) should be small to guarantee the qualitative properties. We determine a threshold mesh size \(h_0\) in terms of the angle condition to ensure the validity of DMPs by Courant FEM.