Sufficient conditions for strong discrete maximum principles in finite element solutions of linear and semilinear elliptic equations
arXiv:2509.00932
The paper presents a new technique to establish global strong discrete maximum principles for finite element solutions of linear and semilinear elliptic equations, even when standard matrix-based conditions fail, by extending macroelement results via a connectivity argument and applying it to problematic meshes.
Abstract
We introduce a novel technique for proving global strong discrete maximum principles for finite element discretizations of linear and semilinear elliptic equations for cases when the common, matrix-based sufficient conditions are not satisfied. The basic argument consists of extending the strong form of discrete maximum principle from macroelements to the entire domain via a connectivity argument. The method is applied to discretizations of elliptic equations with certain pathological meshes, and to semilinear elliptic equations.
40 pages, 14 figures