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Numerical analysis of semi‐linear parabolic systems in diffusion–reaction problems
Author(s) -
Fayyad D.,
Nassif N.
Publication year - 1994
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.1670170105
Subject(s) - mathematics , reaction–diffusion system , bounded function , norm (philosophy) , parabolic partial differential equation , mathematical analysis , partial differential equation , singularity , order (exchange) , type (biology) , finite element method , thermodynamics , ecology , physics , finance , political science , law , economics , biology
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 ).