Well-posedness and discrete analysis for advection-diffusion-reaction in poroelastic media

TL;DR

This paper establishes the well-posedness of a complex PDE system modeling poromechanical and chemical processes in media, and proposes a stable mixed finite element method with verified convergence through numerical experiments.

Contribution

It provides the first rigorous analysis of the well-posedness and stability of a coupled poromechanical and chemical PDE system, along with a validated finite element scheme.

Findings

Proved well-posedness of the nonlinear PDE system.

Developed a stable mixed finite element method.

Numerical experiments confirmed theoretical convergence rates.

Abstract

We analyse a PDE system modelling poromechanical processes (formulated in mixed form using the solid deformation, fluid pressure, and total pressure) interacting with diffusing and reacting solutes in the medium. We investigate the well-posedness of the nonlinear set of equations using fixed-point theory, Fredholm's alternative, a priori estimates, and compactness arguments. We also propose a mixed finite element method and rigorously demonstrate the stability of the scheme. Error estimates are derived in suitable norms, and numerical experiments are conducted to illustrate the mechano-chemical coupling and to verify the theoretical rates of convergence.