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We define a finite element method for the coupling of Stokes and nonlinear Poisson--Boltzmann equations. Using Banach contracting principle, the Babu\v{s}ka--Brezzi theory, and the Minty--Browder theorem, we show that the governing equations have a unique weak solution. We also show that the discrete problem is well-posed, establish C\'ea estimates, and derive convergence rates. We exemplify the properties of the proposed scheme via some numerical experiments showcasing convergence and applicability in the study of electro-osmotic flows in micro-channels.

We develop a family of mixed finite element methods for a model of nonlinear poroelasticity where, thanks to a rewriting of the constitutive equations, the permeability depends on the total poroelastic stress and on the fluid pressure and therefore we can use the Hellinger--Reissner principle with weakly imposed stress symmetry for Biot's equations. The problem is adequately structured into a coupled system consisting of one saddle-point formulation, one linearised perturbed saddle-point formulation, and two off-diagonal perturbations. This system's unique solvability requires assumptions on regularity and Lipschitz continuity of the inverse permeability, and the analysis follows fixed-point arguments and the Babu\v{s}ka--Brezzi theory. The discrete problem is shown uniquely solvable by applying similar fixed-point and saddle-point techniques as for the continuous case. The method is based on the classical PEERS$_k$ elements, it is exactly momentum and mass conservative, and it is robust with respect to the nearly incompressible as well as vanishing storativity limits. We derive a priori error estimates, we also propose fully computable residual-based a posteriori error indicators, and show that they are reliable and efficient with respect to the natural norms, and robust in the limit of near incompressibility. These a posteriori error estimates are used to drive adaptive mesh refinement. The theoretical analysis is supported and illustrated by several numerical examples in 2D and 3D.

We propose a model for the coupling between free fluid and a linearized poro-hyperelastic body. In this model, the Brinkman equation is employed for fluid flow in the porous medium, incorporating inertial effects into the fluid dynamics. A generalized poromechanical framework is used, incorporating fluid inertial effects in accordance with thermodynamic principles. We carry out the analysis of the unique solvability of the governing equations, and the existence proof relies on an auxiliary multi-valued parabolic problem. We propose a Lagrange multiplier-based mixed finite element method for its numerical approximation and show the well-posedness of both semi-and fully-discrete problems. Then, a priori error estimates for both the semi- and fully-discrete schemes are derived. A series of numerical experiments is presented to confirm the theoretical convergence rates, and we also employ the proposed monolithic scheme to simulate 2D physical phenomena in geophysical fluids and biomechanics of the brain function.

This paper discusses lowest-order nonstandard finite element methods for space discretization and explicit and implicit schemes for time discretization of the biharmonic wave equation with clamped boundary conditions. A modified Ritz projection operator defined on $H^2_0(\Omega)$ ensures error estimates under appropriate regularity assumptions on the solution. Stability results and error estimates of optimal order are established in suitable norms for the semidiscrete and explicit/implicit fully-discrete versions of the proposed schemes. Finally, we report on numerical experiments using explicit and implicit schemes for time discretization and Morley, discontinuous Galerkin, and {C$^0$ interior} penalty schemes for space discretization, that validate the theoretical error estimates.

In this paper we propose and analyse a new formulation and pointwise divergence-free mixed finite element methods for the numerical approximation of Darcy--Brinkman equations in vorticity--velocity--pressure form, coupled with a transport equation for thermal energy with viscous dissipative effect and mixed Navier-type boundary conditions. The solvability analysis of the continuous and discrete problems is significantly more involved than usual as it hinges on Banach spaces needed to properly control the advective and dissipative terms in the non-isothermal energy balance equation. We proceed by decoupling the set of equations and use the Banach fixed-point theorem in combination with the abstract theory for perturbed saddle-point problems. Some of the necessary estimates are straightforward modifications of well-known results, while other technical tools require a more elaborated analysis. The velocity is approximated by Raviart--Thomas elements, the vorticity uses N\'ed\'elec spaces of the first kind, the pressure is approximated by piecewise polynomials, and the temperature by continuous and piecewise polynomials of one degree higher than pressure. Special care is needed to establish discrete inf-sup conditions since the curl of the discrete vorticity is not necessarily contained in the discrete velocity space, therefore suggesting to use two different Raviart--Thomas interpolants. A discrete fixed-point argument is used to show well-posedness of the Galerkin scheme. Error estimates in appropriate norms are derived, and a few representative numerical examples in 2D and 3D and with mixed boundary conditions are provided.

We propose a model for the coupling of flow and transport equations with porous membrane-type conditions on part of the boundary. The governing equations consist of the incompressible Navier--Stokes equations coupled with an advection-diffusion equation, and we employ a Lagrange multiplier to enforce the coupling between penetration velocity and transport on the membrane, while mixed boundary conditions are considered in the remainder of the boundary. We show existence and uniqueness of the continuous problem using a fixed-point argument. Next, an H(div)-conforming finite element formulation is proposed, and we address its a priori error analysis. The method uses an upwind approach that provides stability in the convection-dominated regime. We showcase a set of numerical examples validating the theory and illustrating the use of the new methods in the simulation of reverse osmosis processes.

This paper aims first to perform robust continuous analysis of a mixed nonlinear formulation for stress-assisted diffusion of a solute that interacts with an elastic material, and second to propose and analyse a virtual element formulation of the model problem. The two-way coupling mechanisms between the Herrmann formulation for linear elasticity and the reaction-diffusion equation (written in mixed form) consist of diffusion-induced active stress and stress-dependent diffusion. The two sub-problems are analysed using the extended Babu\v{s}ka--Brezzi--Braess theory for perturbed saddle-point problems. The well-posedness of the nonlinearly coupled system is established using a Banach fixed-point strategy under the smallness assumption on data. The virtual element formulations for the uncoupled sub-problems are proven uniquely solvable by a fixed-point argument in conjunction with appropriate projection operators. We derive the a priori error estimates, and test the accuracy and performance of the proposed method through computational simulations.

In this paper, we propose a new formulation and a suitable finite element method for the steady coupling of viscous flow in deformable porous media using divergence-conforming filtration fluxes. The proposed method is based on the use of parameter-weighted spaces, which allows for a more accurate and robust analysis of the continuous and discrete problems. Furthermore, we conduct a solvability analysis of the proposed method and derive optimal error estimates in appropriate norms. These error estimates are shown to be robust in the case of large Lam\'e parameters and small permeability and storativity coefficients. To illustrate the effectiveness of the proposed method, we provide a few representative numerical examples, including convergence verification, poroelastic channel flow simulation, and test the robustness of block-diagonal preconditioners with respect to model parameters.

We propose four-field and five-field Hu--Washizu-type mixed formulations for nonlinear poroelasticity -- a coupled fluid diffusion and solid deformation process -- considering that the permeability depends on a linear combination between fluid pressure and dilation. As the determination of the physical strains is necessary, the first formulation is written in terms of the primal unknowns of solid displacement and pore fluid pressure as well as the poroelastic stress and the infinitesimal strain, and it considers strongly symmetric Cauchy stresses. The second formulation imposes stress symmetry in a weak sense and it requires the additional unknown of solid rotation tensor. We study the unique solvability of the problem using the Banach fixed-point theory, properties of twofold saddle-point problems, and the Banach--Ne\v{c}as--Babu\v{s}ka theory. We propose monolithic Galerkin discretisations based on conforming Arnold--Winther for poroelastic stress and displacement, and either PEERS or Arnold--Falk--Winther finite element families for the stress-displacement-rotation field variables. The wellposedness of the discrete problem is established as well, and we show a priori error estimates in the natural norms. Some numerical examples are provided to confirm the rates of convergence predicted by the theory, and we also illustrate the use of the formulation in some typical tests in Biot poroelasticity.

This paper analyses conforming and nonconforming virtual element formulations of arbitrary polynomial degrees on general polygonal meshes for the coupling of solid and fluid phases in deformable porous plates. The governing equations consist of one fourth-order equation for the transverse displacement of the middle surface coupled with a second-order equation for the pressure head relative to the solid with mixed boundary conditions. We propose novel enrichment operators that connect nonconforming virtual element spaces of general degree to continuous Sobolev spaces. These operators satisfy additional orthogonal and best-approximation properties (referred to as a conforming companion operator in the context of finite element methods), which play an important role in the nonconforming methods. This paper proves a priori error estimates in the best-approximation form, and derives residual--based reliable and efficient a posteriori error estimates in appropriate norms, and shows that these error bounds are robust with respect to the main model parameters. The computational examples illustrate the numerical behaviour of the suggested virtual element discretisations and confirm the theoretical findings on different polygonal meshes with mixed boundary conditions.

We propose, analyze and implement a virtual element discretization for an interfacial poroelasticity-elasticity consolidation problem. The formulation of the time-dependent poroelasticity equations uses displacement, fluid pressure, and total pressure, and the elasticity equations are written in the displacement-pressure formulation. The construction of the virtual element scheme does not require Lagrange multipliers to impose the transmission conditions (continuity of displacement and total traction, and no-flux for the fluid) on the interface. We show the stability and convergence of the virtual element method for different polynomial degrees, and the error bounds are robust with respect to delicate model parameters (such as Lame constants, permeability, and storativity coefficient). Finally, we provide numerical examples that illustrate the properties of the scheme.

We propose a new mathematical model for the interaction of skin cell populations with fibroblast growth factor and bone morphogenetic protein, occurring within deformable porous media. The equations for feather primordia pattering are based on the work by K.J. Painter et al. [J. Theoret. Biol., 437 (2018) 225--238]. We perform a linear stability analysis to identify relevant parameters in the coupling mechanisms, focusing in the regime of infinitesimal strains. We also extend the model to the case of nonlinear poroelasticity and include solid growth by means of Lee decompositions of the deformation gradient. We present a few illustrative computational examples in 2D and 3D, and briefly discuss the design of tailored efficient solvers., Comment: 25 pages, 9 figures

In this article, we propose and analyze a fully coupled, nonlinear, and energy-stable virtual element method (VEM) for solving the coupled Poisson-Nernst-Planck (PNP) and Navier--Stokes (NS) equations modeling microfluidic and electrochemical systems (diffuse transport of charged species within incompressible fluids coupled through electrostatic forces). A mixed VEM is employed to discretize the NS equations whereas classical VEM in primal form is used to discretize the PNP equations. The stability, existence and uniqueness of solution of the associated VEM are proved by fixed point theory. Global mass conservation and electric energy decay of the scheme are also proved. Also, we obtain unconditionally optimal error estimates for both the electrostatic potential and ionic concentrations of PNP equations in the $H^{1}$-norm, as well as for the velocity and pressure of NS equations in the $\mathbf{H}^{1}$- and $L^{2}$-norms, respectively. Finally, several numerical experiments are presented to support the theoretical analysis of convergence and to illustrate the satisfactory performance of the method in simulating the onset of electrokinetic instabilities in ionic fluids, and studying how they are influenced by different values of ion concentration and applied voltage. These tests relate to applications in the desalination of water.

A strongly symmetric stress approximation is proposed for the Brinkman equations with mixed boundary conditions. The resulting formulation solves for the Cauchy stress using a symmetric interior penalty discontinuous Galerkin method. Pressure and velocity are readily post-processed from stress, and a second post-process is shown to produce exactly divergence-free discrete velocities. We demonstrate the stability of the method with respect to a DG-energy norm and obtain error estimates that are explicit with respect to the coefficients of the problem. We derive optimal rates of convergence for the stress and for the post-processed variables. Moreover, under appropriate assumptions on the mesh, we prove optimal $L^2$-error estimates for the stress. Finally, we provide numerical examples in 2D and 3D., Comment: 17 pages

We propose and rigorously analyse semi- and fully discrete discontinuous Galerkin methods for an initial and boundary value problem describing inertial viscoelasticity in terms of elastic and viscoelastic stress components, and with mixed boundary conditions. The arbitrary-order spatial discretisation imposes strongly the symmetry of the stress tensor, and it is combined with a Newmark trapezoidal rule as time-advancing scheme. We establish stability and convergence properties, and the theoretical findings are further confirmed via illustrative numerical simulations in 2D and 3D., Comment: 24 pages

We propose and analyse an augmented mixed finite element method for the Navier--Stokes equations written in terms of velocity, vorticity, and pressure with non-constant viscosity and no-slip boundary conditions. The weak formulation includes least-squares terms arising from the constitutive equation and from the incompressibility condition. The theoretical and practical implications of using augmentation is discussed in detail. In addition, we use fixed--point strategies to show the existence and uniqueness of continuous and discrete solutions under the assumption of sufficiently small data. The method is constructed using any compatible finite element pair for velocity and pressure as dictated by Stokes inf-sup stability, while for vorticity any generic discrete space (of arbitrary order) can be used. We establish optimal a priori error estimates. Finally, we provide a set of numerical tests in 2D and 3D illustrating the behaviour of the scheme as well as verifying the theoretical convergence rates.

In this paper we advance the analysis of discretizations for a fluid-structure interaction model of the monolithic coupling between the free flow of a viscous Newtonian fluid and a deformable porous medium separated by an interface. A five-field mixed-primal finite element scheme is proposed solving for Stokes velocity-pressure and Biot displacement-total pressure-fluid pressure. Adequate inf-sup conditions are derived, and one of the distinctive features of the formulation is that its stability is established robustly in all material parameters. We propose robust preconditioners for this perturbed saddle-point problem using appropriately weighted operators in fractional Sobolev and metric spaces at the interface. The performance is corroborated by several test cases, including the application to interfacial flow in the brain.

In this paper we propose a novel coupled poroelasticity-diffusion model for the formation of extracellular oedema and infectious myocarditis valid in large deformations, manifested as an interaction between interstitial flow and the immune-driven dynamics between leukocytes and pathogens. The governing partial differential equations are formulated in terms of skeleton displacement, fluid pressure, Lagrangian porosity, and the concentrations of pathogens and leukocytes. A five-field finite element scheme is proposed for the numerical approximation of the problem, and we provide the stability analysis for a simplified system emanating from linearisation. We also discuss the construction of an adequate, Schur complement based, nested preconditioner. The produced computational tests exemplify the properties of the new model and of the finite element schemes., Comment: 29 pages

In embryogenesis, epithelial cells, acting as individual entities or as coordinated aggregates in a tissue, exhibit strong coupling between chemical signalling and mechanical responses to internally or externally applied stresses. Intercellular communication in combination with such coordination of morphogenetic movements can lead to drastic modifications in the calcium distribution in the cells. In this paper we extend the recent mechanochemical model in [K. Kaouri, P.K. Maini, P.A. Skourides, N. Christodoulou, S.J. Chapman. J. Math. Biol., 78 (2019) 2059--2092], 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 mixed finite element method 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. We investigate numerically the solutions of this system and show that solitary waves and periodic wavetrains of calcium propagate through the embryonic epithelial sheet. We analyse the bifurcations of the system guided by the bifurcation analysis of the one-dimensional model. We also demonstrate the nucleation of calcium sparks into synchronous calcium waves coupled with contraction. This coupled 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., Comment: 29 pages

We introduce a stress/total-pressure formulation for poroelasticity that includes the coupling with steady nonlinear diffusion modified by stress. The nonlinear problem is written in mixed-primal form, coupling a perturbed twofold saddle-point system with an elliptic problem. The continuous formulation is analysed in the framework of abstract fixed-point theory and Fredholm alternative for compact operators. A mixed finite element method is proposed and its stability and convergence analysis are carried out. We also include a few illustrative numerical tests. The resulting model can be used to study waste removal in the brain parenchyma, where diffusion of a tracer alone or a combination of advection and diffusion are not sufficient to explain the alterations in rates of filtration observed in porous media samples.

We develop the \textit{a posteriori} error analysis of three mixed finite element formulations for rotation-based equations in elasticity, poroelasticity, and interfacial elasticity-poroelasticity. The discretisations use $H^1$-conforming finite elements of degree $k+1$ for displacement and fluid pressure, and discontinuous piecewise polynomials of degree $k$ for rotation vector, total pressure, and elastic pressure. Residual-based estimators are constructed, and upper and lower bounds (up to data oscillations) for all global estimators are rigorously derived. The methods are all robust with respect to the model parameters (in particular, the Lam\'e constants), they are valid in 2D and 3D, and also for arbitrary polynomial degree $k\geq 0$. The error behaviour predicted by the theoretical analysis is then demonstrated numerically on a set of computational examples including different geometries on which we perform adaptive mesh refinement guided by the \textit{a posteriori} error estimators.

We consider a multiphysics model for the flow of Newtonian fluid coupled with Biot consolidation equations through an interface, and incorporating total pressure as an unknown in the poroelastic region. A new mixed-primal finite element scheme is proposed solving for the pairs fluid velocity - pressure and displacement - total poroelastic pressure using Stokes-stable elements, and where the formulation does not require Lagrange multipliers to set up the usual transmission conditions on the interface. The stability and well-posedness of the continuous and semi-discrete problems are analysed in detail. Our numerical study is framed in the context of applicative problems pertaining to heterogeneous geophysical flows and to eye poromechanics. For the latter, we investigate different interfacial flow regimes in Cartesian and axisymmetric coordinates that could eventually help describe early morphologic changes associated with glaucoma development in canine species., Comment: 36 pages

We investigate the behaviour of flux-driven flow through a single-phase fluid domain coupled to a biphasic poroelastic domain. The fluid domain consists of an incompressible Newtonian viscous fluid while the poroelastic domain consists of a linearly elastic solid filled with the same viscous fluid. The material properties of the poroelastic domain, i.e. permeability and elastic parameters, depend on the inhomogeneous initial porosity field. We identify the dimensionless parameters governing the behaviour of the coupled problem: the ratio between the magnitudes of the driving velocity and the Darcy flows in the poroelastic domain, and the ratio between the viscous pressure scale and the size of the elastic stresses in the poroelastic domain. We consider a perfusion system, where flow is forced to pass from the single-phase fluid to the biphasic poroelastic domain. We focus on a simplified two dimensional geometry with small aspect ratio, and perform an asymptotic analysis to derive analytical solutions. The slender geometry is divided in four regions, two outer domains that describe the regions away from the interface and two inner domains that are the regions across the interface. Our analysis advances the quantitative understanding of the role of heterogeneous material properties of a poroelastic domain on its mechanical response when coupled with a fluid domain. The analysis reveals that, in the interfacial zone, the fluid and the elastic behaviours of this coupled Stokes - poroelastic problem can be treated separately via (i) a Stokes - Darcy coupling and (ii) the solid skeleton being stress free.

The analysis of the double-diffusion model and $\mathbf{H}(\mathrm{div})$-conforming method introduced in [B\"urger, M\'endez, Ruiz-Baier, SINUM (2019), 57:1318--1343] is extended to the time-dependent case. In addition, the efficiency and reliability analysis of residual-based {\it a posteriori} error estimators for the steady, semi-discrete, and fully discrete problems is established. The resulting methods are applied to simulate the sedimentation of small particles in salinity-driven flows. The method consists of Brezzi-Douglas-Marini approximations for velocity and compatible piecewise discontinuous pressures, whereas Lagrangian elements are used for concentration and salinity distribution. Numerical tests confirm the properties of the proposed family of schemes and of the adaptive strategy guided by the {\it a posteriori} error indicators., Comment: 30 pages

A variational formulation is introduced for the Oseen equations written in terms of vor\-ti\-city and Bernoulli pressure. The velocity is fully decoupled using the momentum balance equation, and it is later recovered by a post-process. A finite element method is also proposed, consisting in equal-order N\'ed\'elec finite elements and piecewise continuous polynomials for the vorticity and the Bernoulli pressure, respectively. The {\it a priori} error analysis is carried out in the $\mathrm{L}^2$-norm for vorticity, pressure, and velocity; under a smallness assumption either on the convecting velocity, or on the mesh parameter. Furthermore, an {\it a posteriori} error estimator is designed and its robustness and efficiency are studied using weighted norms. Finally, a set of numerical examples in 2D and 3D is given, where the error indicator serves to guide adaptive mesh refinement. These tests illustrate the behaviour of the new formulation in typical flow conditions, and they also confirm the theoretical findings., Comment: 24 pages

We propose and analyse an augmented mixed finite element method for the Oseen equations written in terms of velocity, vorticity, and pressure with non-constant viscosity and homogeneous Dirichlet boundary condition for the velocity. The weak formulation includes least-squares terms arising from the constitutive equation and from the incompressibility condition, and we show that it satisfies the hypotheses of the Babu\vska-Brezzi theory. Repeating the arguments of the continuous analysis, the stability and solvability of the discrete problem are established. The method is suited for any Stokes inf-sup stable finite element pair for velocity and pressure, while for vorticity any generic discrete space (of arbitrary order) can be used. A priori and a posteriori error estimates are derived using two specific families of discrete subspaces. Finally, we provide a set of numerical tests illustrating the behaviour of the scheme, verifying the theoretical convergence rates, and showing the performance of the adaptive algorithm guided by residual a posteriori error estimation., Comment: 18 pages

In this work we address the analysis of the stationary generalized Burgers-Huxley equation (a nonlinear elliptic problem with anomalous advection) and propose conforming, nonconforming and discontinuous Galerkin finite element methods for its numerical approximation. The existence, uniqueness and regularity of weak solutions is discussed in detail using a Faedo-Galerkin approach and fixed-point theory, and a priori error estimates for all three types of numerical schemes are rigorously derived. A set of computational results are presented to show the efficacy of the proposed methods.

We develop robust solvers for a class of perturbed saddle-point problems arising in the study of a second-order elliptic equation in mixed form (in terms of flux and potential), and of the four-field formulation of Biot's consolidation problem for linear poroelasticity (using displacement, filtration flux, total pressure and fluid pressure). The stability of the continuous variational mixed problems, which hinges upon using adequately weighted spaces, is addressed in detail; and the efficacy of the proposed preconditioners, as well as their robustness with respect to relevant material properties, is demonstrated through several numerical experiments.

We perform the linear stability analysis for a new model for poromechanical processes with inertia (formulated in mixed form using the solid deformation, fluid pressure, and total pressure) interacting with diffusing and reacting solutes convected in the medium. We find parameter regions that lead to spatio-temporal instabilities of the coupled system. The mutual dependences between deformation and diffusive patterns are of substantial relevance in the study of morphoelastic changes in biomaterials. We provide a set of computational examples in 2D and 3D (related to brain mechanobiology) that can be used to form a better understanding on how, and up to which extent, the deformations of the porous structure dictate the generation and suppression of spatial patterning dynamics, also related to the onset of mechano-chemical waves.

A virtual element discretisation for the numerical approximation of the three-field formulation of linear poroelasticity introduced in [R. Oyarz\'ua and R. Ruiz-Baier, Locking-free finite element methods for poroelasticity, SIAM J. Numer. Anal. 54 (2016) 2951--2973] is proposed. The treatment is extended to include also the transient case. Appropriate poroelasticity projector operators are introduced and they assist in deriving energy bounds for the time-dependent discrete problem. Under standard assumptions on the computational domain, optimal a priori error estimates are established. Furthermore, the accuracy of the method is verified numerically through a set of computational tests.

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 rigorously 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.

In this brief note, we introduce a non-symmetric mixed finite element formulation for Brinkman equations written in terms of velocity, vorticity and pressure with non-constant viscosity. The analysis is performed by the classical Babu\v{s}ka-Brezzi theory, and we state that any inf-sup stable finite element pair for Stokes approximating velocity and pressure can be coupled with a generic discrete space of arbitrary order for the vorticity. We establish optimal a priori error estimates which are further confirmed through computational examples

We propose and analyse the properties of a new class of models for the electromechanics of cardiac tissue. The set of governing equations consists of nonlinear elasticity using a viscoelastic and orthotropic exponential constitutive law (this is so for both active stress and active strain formulations of active mechanics) coupled with a four-variable phenomenological model for human cardiac cell electrophysiology, which produces an accurate description of the action potential. The conductivities in the model of electric propagation are modified according to stress, inducing an additional degree of nonlinearity and anisotropy in the coupling mechanisms; and the activation model assumes a simplified stretch-calcium interaction generating active tension or active strain. The influence of the new terms in the electromechanical model is evaluated through a sensitivity analysis, and we provide numerical validation through a set of computational tests using a novel mixed-primal finite element scheme.

We propose a new model to describe diffusion processes within active deformable media. Our general theoretical framework is based on physical and mathematical considerations, and it suggests to use diffusion tensors directly coupled to mechanical stress. A proof-of-concept experiment and the proposed generalised reaction-diffusion-mechanics model reveal that initially isotropic and homogeneous diffusion tensors turn into inhomogeneous and anisotropic quantities due to the intrinsic structure of the nonlinear coupling. We study the physical properties leading to these effects, and investigate mathematical conditions for its occurrence. Together, the experiment, the model, and the numerical results obtained using a mixed-primal finite element method, clearly support relevant consequences of stress-assisted diffusion into anisotropy patterns, drifting, and conduction velocity of the resulting excitation waves. Our findings also indicate the applicability of this novel approach in the description of mechano-electrical feedback in actively deforming bio-materials such as the heart.