In embryogenesis, epithelial cells acting as individual entities or as coordinated aggregates in a tissue, exhibit strong coupling between mechanical responses to internally or externally applied stresses and chemical signalling. One of the most important chemical signals in this process is calcium. This mechanochemical coupling and intercellular communication drive the coordination of morphogenetic movements which are characterised by drastic changes in the concentration of calcium in the tissue. In this paper we extend the recent mechanochemical model in Kaouri et al. (J. Math. Biol. 78, 2059–2092, 2019), for an epithelial continuum in one dimension, to a more realistic multi-dimensional case. The resulting parametrised governing equations consist of an advection-diffusion-reaction system for calcium signalling coupled with active-stress linear viscoelasticity and equipped with pure Neumann boundary conditions. We implement a finite element method in perturbed saddle-point form for the simulation of this complex multiphysics problem. Special care is taken in the treatment of the stress-free boundary conditions for the viscoelasticity in order to eliminate rigid motions from the space of admissible displacements. The stability and solvability of the continuous weak formulation is shown using fixed-point theory. Guided by the bifurcation analysis of the one-dimensional model, we analyse the behaviour of the system as two bifurcation parameters vary: the level of IP3 concentration and the strength of the mechanochemical coupling. We identify the parameter regions giving rise to solitary waves and periodic wavetrains of calcium. Furthermore, we demonstrate the nucleation of calcium sparks into synchronous calcium waves coupled with deformation. This model can be employed to gain insights into recent experimental observations in the context of embryogenesis, but also in other biological systems such as cancer cells, wound healing, keratinocytes, or white blood cells.

A stationary Navier-Stokes-Brinkman model coupled to a system of advectiondiffusion equations serves as a model for so-called double-diffusive viscous flow in porous media in which both heat and a solute within the fluid phase are subject to transport and diffusion. The solvability analysis of these governing equations results as a combination of compactness arguments and fixed-point theory. In addition an H(div)-conforming discretisation is formulated by a modification of existing methods for Brinkman flows. The well-posedness of the discrete Galerkin formulation is also discussed, and convergence properties are derived rigorously. Computational tests confirm the predicted rates of error decay and illustrate the applicability of the methods for the simulation of bacterial bioconvection and thermohaline circulation problems.

In this paper, we extend a dual-mixed formulation for a nonlinear generalized Stokes problem to a Huber regularization of the Herschel-Bulkey flow problem. The present approach is based on a two-fold saddle point nonlinear operator equation for the corresponding weak formulation. We provide the uniqueness of solutions for the continuous formulation and propose a discrete scheme based on Arnold-Falk-Winther finite elements. The discretization scheme yields a system of Newton differentiable nonlinear equations, for which a semismooth Newton algorithm is proposed and implemented. Local superlinear convergence of the method is also proved. Finally, we perform several numerical experiments to investigate the behavior and efficiency of the method.

BioOne Complete (complete.BioOne.org) is a full-text database of 200 subscribed and open-access titles in the biological, ecological, and environmental sciences published by nonprofit societies, associations, museums, institutions, and presses.

Entropy stable schemes for the numerical solution of initial value problems of nonlinear, possibly strongly degenerate systems of convection–diffusion equations were recently proposed in Jerez and Parés's study. These schemes extend the theoretical framework of Tadmor's study to convection–diffusion systems. They arise from entropy conservative schemes by adding a small amount of viscosity to avoid spurious oscillations. The main condition for feasibility of entropy conservative or stable schemes for a given model is that the corresponding first‐order system of conservation laws possesses a convex entropy function and corresponding entropy flux, and that the diffusion matrix multiplied by the inverse of the Hessian of the entropy is positive semidefinite. As a new contribution, it is demonstrated in the present work, first, that these schemes can naturally be extended to initial‐boundary value problems with zero‐flux boundary conditions in one space dimension, including an explicit bound on the growth of the total entropy. Second, it is shown that these assumptions are satisfied by certain diffusively corrected multiclass kinematic flow models of arbitrary size that describe traffic flow or the settling of dispersions and emulsions, where the latter application gives rise to zero‐flux boundary conditions. Numerical examples illustrate the behavior and accuracy of entropy stable schemes for these applications.