Multiscale Galerkin approximation scheme for a system of quasilinear parabolic equations

TL;DR

This paper introduces a multiscale Galerkin approximation for coupled quasilinear parabolic equations modeling porous media combustion, demonstrating convergence and potential for accurate macroscopic predictions in relevant temperature regimes.

Contribution

It develops a novel multiscale Galerkin scheme for a complex coupled system derived from pore-scale models, including nonlinear reaction effects, and proves its convergence.

Findings

Numerical simulations show the model's ability to replicate porous media combustion behavior.

The scheme provides uniform estimates ensuring convergence of finite-dimensional approximations.

Distinctions between microscopic reaction effects and pure diffusion are highlighted.

Abstract

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.