Primal-mixed formulations for reaction–diffusion systems on deforming domains

Abstract

We propose a finite element formulation for a coupled elasticity–reaction–diffusion system written in a fully Lagrangian form and governing the spatio-temporal interaction of species inside an elastic, or hyper-elastic body. A primal weak formulation is the baseline model for the reaction–diffusion system written in the deformed domain, and a finite element method with piecewise linear approximations is employed for its spatial discretization. On the other hand, the strain is introduced as mixed variable in the equations of elastodynamics, which in turn acts as coupling field needed to update the diffusion tensor of the modified reaction–diffusion system written in a deformed domain. The discrete mechanical problem yields a mixed finite element scheme based on row-wise Raviart–Thomas elements for stresses, Brezzi–Douglas–Marini elements for displacements, and piecewise constant pressure approximations. The application of the present framework in the study of several coupled biological systems on deforming geometries in two and three spatial dimensions is discussed, and some illustrative examples are provided and extensively analyzed.