Numerical analysis of semi-linear parabolic systems in diffusion–reaction problems

In this paper, we investigate problems of approximation for the solution of a system of coupled semi-linear parabolic partial differential equations that model diffusion-reaction problems in chemical engineering. Given that the solutions belong to H^s (0, ∞), we consider finite-element approximations on bounded domains (0, R(h)) such that lim_h→0R(h)] = ∞. Optimal convergence estimates are found to depend on the asymptotic behaviour of the solution and its regularity near t = 0. In the L²-norm, they cannot exceed an order of O((;h²/t^3/4) + h²In h]²). For that purpose, a Wheeler-type argument is also generalized to non-coercive elliptic forms. Fully discrete schemes that preserve the positivity of the solutions are also considered. Due to the singularity at t = 0, they lead to estimates of the order O(τ^1/4 + h²/t^3/4).