A semidiscrete Galerkin scheme for a coupled two-scale elliptic-parabolic system: well-posedness and convergence approximation rates

TL;DR

This paper develops a semidiscrete Galerkin scheme for a coupled elliptic-parabolic system modeling multiscale reactive flow in heterogeneous materials, proving well-posedness, convergence, and providing simulation validation.

Contribution

It introduces a novel two-scale numerical approximation method with proven error estimates for a complex coupled system, ensuring well-posedness and convergence.

Findings

The scheme is well-posed and convergent.

Error estimates are established for the approximation.

Simulations confirm the theoretical results.

Abstract

In this paper, we study the numerical approximation of a coupled system of elliptic-parabolic equations posed on two separated spatial scales. The model equations describe the interplay between macroscopic and microscopic pressures in an unsaturated heterogeneous medium with distributed microstructures as they often arise in modeling reactive flow in cementitious-based materials. Besides ensuring the well-posedness of our two-scale model, we design two-scale convergent numerical approximations and prove a priori error estimates for the semidiscrete case. We complement our analysis with simulation results illustrating the expected behaviour of the system.