Fully mixed finite element methods for the coupling of viscoelasticity and reaction-diffusion models

Abstract

We propose and analyse a fully mixed finite element method for a two-way coupled mechanochemical model of calcium signalling in viscoelastic tissue. The mechanical response follows a Kelvin--Voigt law, with the elastic and viscous stresses, velocity, and spin tensor as unknowns, and the symmetry of the total stress imposed weakly. The reaction-diffusion system for calcium and inactivated inositol triphosphate is also formulated in mixed form, introducing the diffusive flux as an additional unknown. The coupling is bidirectional: calcium generates an active stress, while local dilation feeds back into the calcium dynamics. We establish global existence and uniqueness by reducing the coupled system to an integro-differential problem and subsequently recovering the mixed variables. The semidiscrete scheme is formulated for general conforming finite element spaces and shown to be well posed under a single inf-sup condition. An interpolation operator that commutes with the divergence and preserves the weak symmetry pairing yields optimal a priori error estimates. We specialise the analysis to three families of weakly symmetric finite elements, deriving explicit convergence rates and, for one family, a superconvergent local post-processing of the displacement. Numerical tests in two and three dimensions confirm the predicted convergence rates and illustrate the method through viscoelastic benchmarks and mechanically driven calcium wavetrains.