A study on nonnegativity preservation in finite element approximation of Nagumo-type nonlinear differential equations

TL;DR

This paper investigates conditions under which finite element methods preserve nonnegativity and boundedness in solving Nagumo-type nonlinear differential equations, considering various discretization schemes and reaction term treatments.

Contribution

It provides new conditions on mesh and time step size for nonnegativity preservation in finite element approximations of Nagumo equations with anisotropic diffusion.

Findings

Conditions for mesh and time step size ensuring nonnegativity.

Effects of lumping mass matrix and reaction term on preservation.

Numerical examples confirming theoretical results.

Abstract

Preservation of nonnengativity and boundedness in the finite element solution of Nagumo-type equations with general anisotropic diffusion is studied. Linear finite elements and the backward Euler scheme are used for the spatial and temporal discretization, respectively. An explicit, an implicit, and two hybrid explicit-implicit treatments for the nonlinear reaction term are considered. Conditions for the mesh and the time step size are developed for the numerical solution to preserve nonnegativity and boundedness. The effects of lumping of the mass matrix and the reaction term are also discussed. The analysis shows that the nonlinear reaction term has significant effects on the conditions for both the mesh and the time step size. Numerical examples are given to demonstrate the theoretical findings.