A study on nonnegativity preservation in finite element approximation of Nagumo-type nonlinear differential equations
Provides theoretical guarantees for nonnegativity preservation in finite element simulations of reaction-diffusion systems, which is important for applications in biology and chemistry.
The study derives mesh and time step conditions for preserving nonnegativity and boundedness in finite element solutions of Nagumo-type equations with anisotropic diffusion, using linear elements and backward Euler time stepping. Numerical examples confirm the theoretical conditions.
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.