We consider the analysis and numerical solution of a forward-backward boundary value problem.We provide some motivation, prove existence and uniqueness in a functionclass especially geared to the problem at hand, provide various energyestimates, prove a priori error estimates for the Galerkin method,and show the results of some numerical computations.

One of the current debate about simulating the electrical activity in the heart is thefollowing: Using a realistic anatomical setting, i.e. realisticgeometries, fibres orientations, etc., is it enough to use a simplified 2-variablephenomenological model to reproduce the main characteristics of the cardiac actionpotential propagation, and in what sense is it sufficient? Using a combination ofdimensional and asymptotic analysis, together with the well-known Mitchell − Schaeffermodel, it is shown that it is possible to accurately control (at least locally) thesolution while spatial propagation is involved. In particular, we reduce the set ofparameters by writing the bidomain model in a new nondimensional form. The parameters ofthe bidomain model with Mitchell − Schaeffer ion kinetics are then set and shown to be inone-to-one relation with the main characteristics of the four phases of a propagatedaction potential. Explicit relations are derived using a combination of asymptotic methodsand ansatz. These relations are tested against numerical results. We illustrate how theserelations can be used to recover the time/space scales and speed of the action potentialin various regions of the heart.

We present a model of the full thermo-mechanical evolution of a shape memory body undergoing a uniaxial tensile stress. The well-posedness of the related quasi-static thermo-inelastic problem is addressed by means of hysteresis operators techniques. As a by-product, details on a time-discretization of the problem are provided.

The swelling and shrinkage of biological tissues are modelled by a four-component mixture theory [J.M. Huyghe and J.D. Janssen, Int. J. Engng. Sci.35 (1997) 793–802; K. Malakpoor, E.F. Kaasschieter and J.M. Huyghe, Mathematical modelling and numerical solution of swelling of cartilaginous tissues. Part I: Modeling of incompressible charged porous media.ESAIM: M2AN41 (2007) 661–678]. This theory results in a coupled system of nonlinear parabolic differential equations together with an algebraic constraint for electroneutrality. In this model, it is desirable to obtain accurate approximations of the fluid flow and ions flow. Such accurate approximations can be determined by the mixed finite element method. The solid displacement, fluid and ions flow and electro-chemical potentials are taken as degrees of freedom. In this article the lowest-order mixed method is discussed. This results into a first-order nonlinear algebraic equation with an indefinite coefficient matrix. The hybridization technique is then used to reduce the list of degrees of freedom and to speed up the numerical computation. The mixed hybrid finite element method is then validated for small deformations using the analytical solutions for one-dimensional confined consolidation and swelling. Two-dimensional results are shown in a swelling cylindrical hydrogel sample.

In this paper, we are interested in modelling the flow of the coolant (water) in anuclear reactor core. To this end, we use a monodimensional low Mach number modelsupplemented with the stiffened gas law. We take into account potential phase transitionsby a single equation of state which describes both pure and mixture phases. In someparticular cases, we give analytical steady and/or unsteady solutions which providequalitative information about the flow. In the second part of the paper, we introduce twovariants of a numerical scheme based on the method of characteristics to simulate thismodel. We study and verify numerically the properties of these schemes. We finally presentnumerical simulations of a loss of flow accident (LOFA) induced by a coolant pump tripevent.

The instationary Stokes and Navier−Stokes equations are considered in a simultaneously space-timevariational saddle point formulation, so involving both velocities u and pressure p. For the instationaryStokes problem, it is shown that the corresponding operator is a boundedlyinvertible linear mapping between H1 and H'2, both Hilbertspaces H1 and H2 beingCartesian products of (intersections of) Bochner spaces, or duals of those. Based on theseresults, the operator that corresponds to the Navier−Stokes equations is shown to mapH1 into H'2, with a Fréchetderivative that, at any (u,p) ∈H1, is boundedly invertible. These resultsare essential for the numerical solution of the combined pair of velocities and pressureas function of simultaneously space and time. Such a numerical approach allows for theapplication of (adaptive) approximation from tensor products of spatial and temporal trialspaces, with which the instationary problem can be solved at a computational complexitythat is of the order as for a corresponding stationary problem.

We investigate different asymptotic regimesfor Vlasov equations modeling the evolution of a cloud of particles in a turbulent flow. In one case we obtain a convection or a convection-diffusion effective equation on the concentration of particles. In the second case, the effective model relies on a Vlasov-Fokker-Planck equation.

The swelling and shrinkage of biological tissues are modelled by a four-component mixture theory in which a deformable and charged porous medium is saturated with a fluid with dissolved ions. Four components are defined: solid, liquid, cations and anions. The aim of this paper is the construction of the Lagrangian model of the four-component system. It is shown that, with the choice of Lagrangian description of the solid skeleton, the motion of the other components can be described in terms of Lagrangian initial system of the solid skeleton as well. Such an approach has a particularly important bearing on computer-aided calculations. Balance laws are derived for each component and for the whole mixture. In cooperation of the second law of thermodynamics, the constitutive equations are given. This theory results in a coupled system of nonlinear parabolic differential equations together with an algebraic constraint for electroneutrality. In this model, it is desirable to obtain an accurate approximation of the fluid flow and ions flow. Such an accurate approximation can be determined by the mixed finite element method. Part II is devoted to this task.

From the fundamental laws of elasticity, we write a model for the contact between two membranes and we perform the analysis of the corresponding system of variational inequalities. We propose a finite element discretization of this problem and prove its well-posedness. We also establish a priori and a posteriori error estimates.

A Legendre spectral collocation method is presented for the solutionof the biharmonic Dirichlet problem on a square. The solution andits Laplacian are approximated using the set of basis functions suggestedby Shen, which are linear combinations of Legendre polynomials. A Schurcomplement approach is used to reduce the resulting linear system to oneinvolving the approximation of the Laplacian of the solution on the twovertical sides of the square. The Schur complement system is solved bya preconditioned conjugate gradient method. The total cost of the algorithmis O(N 3). Numerical results demonstrate the spectral convergence of the method.

Singularly perturbed problems often yield solutions with strong directional features,e.g. with boundary layers. Such anisotropic solutions lend themselves to adapted, anisotropic discretizations. The quality of the corresponding numerical solution is a key issue in any computational simulation.To this end we present a new robust error estimator for a singularly perturbed reaction-diffusion problem. In contrast to conventional estimators, our proposal is suitable for anisotropic finite element meshes. The estimator is based on the solution of a local problem, and yields error bounds uniformly in the small perturbation parameter. The error estimation is efficient, i.e. a lower error bound holds. The error estimator is also reliable, i.e. an upper error bound holds, provided that the anisotropic mesh discretizes the problem sufficiently well. A numerical example supports the analysis of our anisotropic error estimator.

We investigate sufficient and possiblynecessary conditions for the L 2 stability of the upwind first orderfinite volume scheme for Maxwell equations, with metallic andabsorbing boundary conditions. We yield a very general sufficient condition,valid for any finite volume partition in two and three spacedimensions. We show this condition is necessary for a class ofregular meshes in two space dimensions. However, numerical tests show it is not necessaryin three space dimensions even on regular meshes. Stability limits for time and space schemes with higher orders of accuracy are numericallyinvestigated.

In this work we introduce an accurate solver for theShallow Water Equations with source terms. This scheme does not need any kind of entropy correction to avoid instabilities near critical points. The scheme also solves the non-homogeneous case, in such a way that all equilibria are computed at least with second order accuracy. We perform several tests for relevant flows showing the performance of our scheme.

In this paper we introduce a new class of numerical schemes for the incompressible Navier-Stokes equations, which are inspired by the theory of discrete kinetic schemes for compressible fluids. For these approximations it is possible to give a stability condition, based on a discrete velocities version of the Boltzmann H-theorem. Numerical tests are performed to investigate their convergence and accuracy.

The aim of this work is to compare a new uncoupled solver for the cardiac Bidomain model with a usual coupled solver. The Bidomain model describes the bioelectric activity of the cardiac tissue and consists of a system of a non-linear parabolic reaction-diffusion partial differential equation (PDE) and an elliptic linear PDE. This system models at macroscopic level the evolution of the transmembrane and extracellular electric potentials of the anisotropic cardiac tissue. The evolution equation is coupled through the non-linear reaction term with a stiff system of ordinary differential equations (ODEs), the so-called membrane model, describing the ionic currents through the cellular membrane. A novel uncoupled solver for the Bidomain system is here introduced, based on solving twice the parabolic PDE and once the elliptic PDE at each time step, and it is compared with a usual coupled solver. Three-dimensional numerical tests have been performed in order to show that the proposed uncoupled method has the same accuracy of the coupled strategy. Parallel numerical tests on structured meshes have also shown that the uncoupled technique is as scalable as the coupled one. Moreover, the conjugate gradient method preconditioned by Multilevel Hybrid Schwarz preconditioners converges faster for the linear systems deriving from the uncoupled method than from the coupled one. Finally, in all parallel numerical tests considered, the uncoupled technique proposed is always about two or three times faster than the coupled approach.

In this paper we construct a model to describe someaspects of the deformation of the central region of the human lung considered as acontinuous elastically deformable medium. To achieve this purpose, we studythe interaction between the pipes composing the tree and the fluid that goes through it. We use a stationary model to determine the deformed radius of each branch. Then, we solve a constrained minimization problem, so as to minimize the viscous (dissipated) energy in the tree. The key feature of our approach is the useof a fixed point theorem in order to find the optimal flow associatedto a deformed tree. We also give some numerical results withinteresting consequences on human lung deformation during expiration, particularlyconcerning the localization of the equal pressure point (EPP).

The aim of this paper is to analyze a formulation of the eddy current problem in terms of a time-primitive of the electric field in a bounded domain with input current intensities or voltage drops as source data. To this end, we introduce a Lagrange multiplier to impose the divergence-free condition in the dielectric domain. Thus, we obtain a time-dependent weak mixed formulation leading to a degenerate parabolic problem which we prove is well-posed. We propose a finite element method for space discretization based on Nédélec edge elements for the main variable and standard finite elements for the Lagrange multiplier, for which we obtain error estimates. Then, we introduce a backward Euler scheme for time discretization and prove error estimates for the fully discrete problem, too. Finally, the method is applied to solve a couple of test problems.

We propose and analyze several finite-element schemes for solving a grade-twofluid model, with atangential boundary condition, in a two-dimensional polygon. The exactproblem is split into ageneralized Stokes problem and a transport equation, in such a way that italways has a solutionwithout restriction on the shape of the domain and on the size of the data.The first scheme usesdivergence-free discrete velocities and a centered discretization of thetransport term, whereas theother schemes use Hood-Taylor discretizations for the velocity andpressure, and either a centered or an upwinddiscretization of the transport term. One facet of our analysis isthat, without restrictions on the data,each scheme has a discrete solution and all discrete solutions convergestrongly to solutions of theexact problem. Furthermore, if the domain is convex and the data satisfycertain conditions, eachscheme satisfies error inequalities that lead to error estimates.

The hydrostatic approximation of the incompressible 3D stationaryNavier-Stokes equations is widely used in oceanography and other applied sciences. It appears through a limit process due to the anisotropy of the domain in use, an ocean, and it is usually studied as such.We consider in this paper an equivalent formulation to this hydrostatic approximation that includes Coriolis force and an additional pressure term that comes from taking into account the pressure in the state equation for the density. It therefore models a slight dependence of the density upon compression terms. We study this model as an independent mathematical object and prove an existence theorem by means of a mixed variational formulation. The proof uses a family of finite element spaces to discretize the problem coupled with a limit process that yields the solution.We finish this paper with an existence and uniqueness result for the evolutionary linear problem associated to this model. This problem includes the same additional pressure term and Coriolis force.

A new set of nonlocal boundary conditions is proposed for the higher modes of the 3D inviscid primitive equations. Numerical schemes using the splitting-up method are proposed for these modes. Numerical simulations of the full nonlinear primitive equations are performed on a nested set of domains, and the results are discussed.