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This paper presents the unsteady Darcy's equations coupled with two nonlinearreaction-diffusion equations, namely this system describes the massconcentration and heat transfer in porous media. The existence and uniquenessof the solution are established for both the variational formulation problemand for its discrete one obtained using spectral discretization. Optimal apriori estimates are given using the Brezzi-Rappaz-Raviart theorem. We concludeby some numerical tests which are in agreement with our theoretical results.

Analysis of backward Euler/Spectral discretization for an evolutionary mass and heat transfer in porous medium