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

Abstract

h i g h l i g h t s• 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. a b s t r a c t 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.

A positive and bounded finite element approximation of the generalized Burgers–Huxley equation

“…From [24], (19) guarantees that the square coefficient matrix A in (18) is an M -matrix. Hence A is invertible, which implies a unique solution u k h to (13).…”

“…where |J| is the determinant of the matrix that appears in the mapping from the reference triangle T to the triangle T ∈ T h (see [24] (4.6)). For the approximation of u k h in (13) Note that under assumption A4 we have that b k ∞ ≤ α and g k ∞ ≤ β( 1−γ 2 ) 2 ≤ β.…”

“…In [24], conditions to ensure that A is an M -matrix [13] were investigated. These conditions involved the following assumptions on the mesh: For every triangle T not adjacent to ∂Ω, there are positive constants c h , C h , c θ , c J and C J such that:…”

“…The following Lemmas will be helpful in the analysis. [24].) For u, u t , u tt in L 2 ((0, T ], L 2 (Ω)),…”

Finite element analysis and approximation of Burgers’‐Fisher equation

“…We will use the following lemma to prove the convergence of fully discrete problem,Lemma If , then Proof See .Theorem Suppose If u k is a solution of (5.3), and . Moreover, let is a solution of (5.2) and , then …”