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

Finite element methods for large-strain poroelasticity/chemotaxis models simulating the formation of myocardial oedema

“…where the R i 's and Ri 's are scalar and vector residuals, and the β i > 0 are additional constants arising from the linearisation of the reaction terms and from the change of variables (4.1). A variant of system (4.4) including advection and nonlinear reaction, has been studied in [71]. We follow that reference and in the remainder of this section we present the solvability, stability, and convergence analysis for the in-time problem associated with (4.2).…”

“…The essential properties of the new model include a formulation in terms of displacement, fluid pressure, and Lagrangian porosity, coupled with concentrations for leukocytes and pathogens. In particular, the poro-hyperelastic compartment of the model can be identified, after linearisation and adequately choosing the initial guess, with the three-field formulation for Biot's poroelasticity from [57,71]. We have proposed a mixed finite element method together with splitting strategies, and have studied their properties in terms of accuracy and iteration count.…”

Stability analysis for a new model of multi-species convection-diffusion-reaction in poroelastic tissue

“…This work is composed by two main contributions. In [1] we have recently explored the well-posedness of the coupled system and have addressed the stability of a mixed finite element discretisation. On the other hand, in the present companion paper we focus more on detailing the physical aspects of the model, on deriving a spectral linear stability analysis, and in providing numerical examples dealing with growth and pattern formation, as well as with an application in traumatic brain injury.…”

“…Even if a large variety of numerical methods exist for producing approximate solutions, appropriate methods (in the sense of being robust with respect to model parameters, being convergent, and replicating key properties of the underlying physico-chemical phenomena) have appeared only recently. As in [13], here we employ a mixed three-field formulation for poroelasticity, and in [1] we have carried out the theoretical analysis of convergence properties of the scheme, tailored for the coupling with a primal formulation for the convection-diffusion system.…”

“…The remainder of this paper is laid out as follows. The governing equations proposed in [1] are recalled in what is left of this section. Then, in Section 2 we perform a linear stability analysis around a steady state with zero solid displacement, constant fluid pressure, and constant solute concentrations.…”