Sufficient conditions for strong discrete maximum principles in finite element solutions of linear and semilinear elliptic equations
This addresses a theoretical gap in numerical analysis for elliptic equations, but appears incremental as it builds on existing maximum principle frameworks.
The paper tackles the problem of proving global strong discrete maximum principles for finite element discretizations of linear and semilinear elliptic equations when standard matrix-based conditions fail, by introducing a novel technique that uses a connectivity argument to extend results from macroelements to the entire domain, and applies it to pathological meshes and semilinear cases.
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.