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

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 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.

KEYWORDS:

• Biot equations
• reaction-diffusion
• mixed finite element scheme
• well-posedness and stability
• numerical experiments and error estimates

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 65M60
• 74F10
• 35K57
• 74L15

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by Comisión Nacional de Investigación Científica y Tecnológica CONICYT-Chile – through Becas Chile (PhD fellowship for foreign students) 21170275 and by the London Mathematical Society [ Scheme 5, Grant 51703]. Department of Science and Technology (DST-SERB), India – through MATRICS grant MTR/2019/000519.