  We discuss a multiscale Galerkin approximation scheme for a system of coupled
quasilinear parabolic equations. These equations arise from the upscaling of a
pore scale filtration combustion model under the assumptions of large
Damkh\"oler number and small P\'eclet number. The upscaled model consists of a
heat diffusion equation and a mass diffusion equation in the bulk of a
macroscopic domain. The associated diffusion tensors are bivariate functions of
temperature and concentration and provide the necessary coupling conditions to
elliptic-type cell problems. These cell problems are characterized by a
reaction-diffusion phenomenon with nonlinear reactions of Arrhenius type at a
gas-solid interface. We discuss the wellposedness of the quasilinear system and
establish uniform estimates for the finite dimensional approximations. Based on
these estimates, the convergence of the approximating sequence is proved. The
results of numerical simulations demonstrate, in suitable temperature regimes,
the potential of solutions of the upscaled model to mimic those from porous
media combustion. Moreover, distinctions are made between the effects of the
microscopic reaction-diffusion processes on the macroscopic system of equations
and a purely diffusion system.
