Preserving nonnegativity of an affine finite element approximation for a convection–diffusion–reaction problem

• We present a monotone FEM scheme for a convection–diffusion–reaction problem in 2, 3D.
• The considered equation does not possess an underlying maximum principle.
• Sufficient conditions are given to ensure the nonnegativity of the approximations.
• Numerical examples confirm the necessity and sufficiency of the conditions.

An affine finite element scheme approximation of a time dependent linear convection–diffusion–reaction problem in 2D and 3D is presented. For these equations which do not satisfy an underlying maximum principle, sufficient conditions are given in terms of the coefficient functions, the computational grid and the discretization parameters to ensure that the nonnegativity property of the true solution is also satisfied by its approximation. Numerical examples are given which confirm the necessity and sufficiency of the discretization conditions to guarantee the nonnegativity of the approximation.