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→0 [ R(h) ] = ∞. Optimal convergence estimates are found to depend on the asymptotic behaviour of the solution and its regularity near t = 0. In the L 2 ‐norm, they cannot exceed an order of O (( ;h 2 / t 3/4 ) + h 2 [In 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 + h 2 / t 3/4 ).