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Preservation of nonnengativity and boundedness in the finite element solutionof Nagumo-type equations with general anisotropic diffusion is studied. Linearfinite elements and the backward Euler scheme are used for the spatial andtemporal discretization, respectively. An explicit, an implicit, and two hybridexplicit-implicit treatments for the nonlinear reaction term are considered.Conditions for the mesh and the time step size are developed for the numericalsolution to preserve nonnegativity and boundedness. The effects of lumping ofthe mass matrix and the reaction term are also discussed. The analysis showsthat the nonlinear reaction term has significant effects on the conditions forboth the mesh and the time step size. Numerical examples are given todemonstrate the theoretical findings.

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