  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.
