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.

We consider higher order mixed finite element methods for the incompressible Stokes or Navier-Stokes equations with Q r -elements for the velocity anddiscontinuous $P_{r-1}$ -elements for the pressure where the orderr can vary from element to elementbetween 2 and a fixed bound $r^*$ .We prove the inf-sup condition uniformly with respect to the meshwidth hon general quadrilateral and hexahedral meshes with hanging nodes.

Electro-muscular disruption (EMD) devices such as TASER M26 andX26 have been used as a less-than-lethal weapon. Such EMD devicesshoot a pair of darts toward an intended target to generate anincapacitating electrical shock. In the use of the EMD device,there have been controversial questions about its safety andeffectiveness. To address these questions, we need to investigatethe distribution of the current density J inside the targetproduced by the EMD device. One approach is to develop acomputational model providing a quantitative and reliable analysisabout the distribution of J. In this paper, we set up amathematical model of a typical EMD shock, bearing in mind that weare aiming to compute the current density distribution inside thehuman body with a pair of inserted darts. The safety issue ofTASER is directly related to the magnitude of |J| at the regionof the darts where the current density J is highlyconcentrated. Hence, fine computation of J near the dart isessential. For such numerical simulations, serious computationaldifficulties are encountered in dealing with the darts having twodifferent very sharp corners, tip of needle and tip of barb. Theboundary of a small fishhook-shaped dart inside a largecomputational domain and the presence of corner singularitiesrequire a very fine mesh leading to a formidable amount ofnumerical computations. To circumvent these difficulties, wedeveloped a multiple point source method of computing J. It hasa potential to provide effective analysis and more accurateestimate of J near fishhook-shaped darts. Numerical experimentsshow that the MPSM is just fit for the study of EMD shocks.

A three-parameter family of Boussinesq type systems in two spacedimensions is considered. These systems approximate thethree-dimensional Euler equations, and consist of three nonlineardispersive wave equations that describe two-way propagation oflong surface waves of small amplitude in ideal fluids over ahorizontal bottom. For a subset of these systems it is proved thattheir Cauchy problem is locally well-posed in suitable Sobolevclasses. Further, a class of these systems is discretized byGalerkin-finite element methods, and error estimates are provedfor the resulting continuous time semidiscretizations. Results ofnumerical experiments are also presented with the aim of studyingproperties of line solitary waves and expanding wave solutions of these systems.

Modeling the kinetics of a precipitation dissolution reaction occurring in a porous medium where diffusion alsotakes place leads to a system of two parabolic equations and one ordinary differential equation coupled with a stiff reaction term. This system is discretized by a finite volume scheme which is suitable for the approximation of thediscontinuous reaction term of unknown sign. Discrete solutions are shown to exist and converge towards a weak solution of the continuous problem. Uniqueness is proved under a Lipschitz condition on the equilibrium gap function. Numerical tests are shown which prove the efficiency of the scheme.

In this work we deal with the numerical solution of aHamilton-Jacobi-Bellman (HJB) equation with infinitely manysolutions. To compute the maximal solution – the optimalcost of the original optimal control problem – we present acomplete discrete method based on the use of some finite elementsand penalization techniques.


The Diffusion Monte Carlo method is devoted to the computation ofelectronic ground-state energies of molecules. In this paper, we focus onimplementations of this method which consist in exploring theconfiguration space with a fixed number of random walkers evolvingaccording to a stochastic differential equation discretized in time. Weallow stochastic reconfigurations of the walkers to reduce thediscrepancy between the weights that they carry. On a simpleone-dimensional example, we prove the convergence of the method for afixed number of reconfigurations when the number of walkers tends to+∞ while the timestep tends to 0. We confirm our theoreticalrates of convergence by numerical experiments. Various resamplingalgorithms are investigated, both theoretically and numerically.


We present families of scalar nonconforming finite elements of arbitraryorder $r\ge 1$ with optimal approximation properties on quadrilaterals andhexahedra. Their vector-valued versions together with a discontinuouspressure approximation of order $r-1$ form inf-sup stable finite element pairsof order r for the Stokes problem. The well-known elements by Rannacherand Turek are recovered in the case r=1. A numerical comparison betweenconforming and nonconforming discretisations will be given. Since higherorder nonconforming discretisation on quadrilaterals and hexahedra have lessunknowns and much less non-zero matrix entries compared to correspondingconforming methods, these methods are attractive for numerical simulations.


We propose and study some new additive, two-level non-overlapping Schwarz preconditioners for the solution of the algebraic linear systems arising from a wide class of discontinuous Galerkin approximations of elliptic problems that have been proposed up to now. In particular, two-level methods for both symmetric and non-symmetric schemes are introduced and some interesting features, which have no analog in the conforming case, are discussed.Both the construction and analysis of the proposed domain decomposition methods are presented in a unified framework. For symmetric schemes, it is shown that the condition number of the preconditioned system is of order O(H/h), where H and h are the mesh sizes of the coarse and fine grids respectively, which are assumed to be nested. For non-symmetric schemes, we show by numerical computations that the Eisenstat et al. [SIAM J. Numer. Anal.20 (1983) 345–357] GMRES convergence theory, generally used in the analysis of Schwarz methods for non-symmetric problems, cannot be applied even if the numerical results show that the GMRES applied to the preconditioned systems converges in a finite number of steps and the proposed preconditioners seem to be scalable.Extensive numerical experiments to validate our theory and to illustrate the performance and robustness of the proposed two-level methods are presented.


Low order edge elements are widely used for electromagnetic field problems. Higher order edge approximations are receiving increasing interest but their definition become rather complex.In this paper we propose a simple definition for Whitney edge elements of polynomial degree higher than one. We give a geometrical localization of all degrees of freedom over particular edges and provide a basis for these elements on simplicial meshes.As for Whitney edge elements of degree one, the basis is expressedonly in terms of the barycentric coordinates of the simplex.

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.

A continuous finite element method to approximate Friedrichs' systems isproposed and analyzed. Stability is achieved by penalizing the jumpsacross meshinterfaces of the normal derivative of some components of the discrete solution. The convergence analysis leads to optimal convergence ratesin the graph norm and suboptimal of order ½ convergence rates inthe L 2-norm. A variant of the method specialized toFriedrichs' systems associated with elliptic PDE's in mixed form andreducing the number of nonzero entries in the stiffness matrix is alsoproposed and analyzed. Finally, numerical results are presented to illustrate thetheoretical analysis.

We present a sparse grid/hyperbolic cross discretization for many-particle problems.It involves the tensor product of a one-particle multilevel basis. Subsequent truncation of the associated series expansion then results in a sparse grid discretization.Here, depending on the norms involved, different variants of sparse grid techniques for many-particle spaces can be derivedthat, in the best case, result in complexities and error estimates which are independent of the number of particles.Furthermore we introduce an additional constraint which gives antisymmetric sparse grids which are suited to fermionic systems. We apply the antisymmetric sparse grid discretization to the electronic Schrödinger equationand compare costs, accuracy, convergence rates and scalability with respect to the number of electrons present in the system.


Motivated by the pricing of American options for baskets weconsider a parabolic variational inequality in a boundedpolyhedral domain $\Omega\subset\mathbb{R}^d$ with a continuous piecewisesmooth obstacle. We formulate a fully discrete method by usingpiecewise linear finite elements in space and the backward Eulermethod in time. We define an a posteriori error estimator and showthat it gives an upper bound for the error inL2(0,T;H1 (Ω)). The error estimator is localized in thesense that the size of the elliptic residual is only relevant inthe approximate non-contact region, and the approximability of theobstacle is only relevant in the approximate contact region. Wealso obtain lower bound results for the space error indicators inthe non-contact region, and for the time error estimator.Numerical results for d=1,2 show that the error estimator decayswith the same rate as the actual error when the space meshsize hand the time step τ tend to zero. Also, the error indicatorscapture the correct behavior of the errors in both the contact andthe non-contact regions.

We rigorously establish the existence of the limithomogeneous constitutive law of a piezoelectric composite made of periodicallyperforated microstructures and whose reference configuration is athin shell with fixed thickness. We deal with an extension of theKoiter shell model in which the three curvilinear coordinates ofthe elastic displacement field and the electric potential arecoupled. By letting the size of themicrostructure going to zero and by using the periodicunfolding method combined with the Korn's inequality in perforateddomains, we obtain the limit model.

The superconsistent collocation method, which is based on acollocation grid different from the one used to represent thesolution, has proven to be very accurate in the resolution ofvarious functional equations. Excellent results can be alsoobtained for what concerns preconditioning. Some analysis andnumerous experiments, regarding the use of finite-differencespreconditioners, for matrices arising from pseudospectralapproximations of advection-diffusion boundary value problems, arepresented and discussed, both in the case of Legendre andChebyshev representation nodes.

The discretisation of the Oseen problem by finite element methods may sufferin general from two shortcomings. First, the discrete inf-sup (Babuška-Brezzi)condition can be violated. Second, spurious oscillationsoccur due to the dominating convection. One way to overcome bothdifficulties is the use of local projection techniques. Studying the local projection method in an abstract setting, we show that the fulfilment of a local inf-sup condition between approximation andprojection spaces allows to construct an interpolation with additional orthogonality properties. Based on this special interpolation, optimal a-priori error estimates are shown with error constants independent of the Reynolds number. Applying the general theory,we extend the results of Braack and Burman for the standard two-level versionof the local projection stabilisation to discretisations of arbitrary order onsimplices, quadrilaterals, and hexahedra. Moreover, our general theory allowsto derive a novel class of local projection stabilisation by enrichment of the approximation spaces. This class of stabilised schemes uses approximation and projection spaces defined on the same mesh and leads tomuch more compact stencils than in the two-level approach. Finally, on simplices, the spectral equivalence of the stabilising terms of the local projection method and the subgrid modelling introduced by Guermond is shown. This clarifies the relation of the local projection stabilisation to the variational multiscaleapproach.

Many numerical simulations in (bilinear) quantum control use the monotonically convergent Krotov algorithms (introduced byTannor et al. [Time Dependent Quantum Molecular Dynamics (1992) 347–360]), Zhu and Rabitz [J. Chem. Phys. (1998) 385–391] or theirunified form described in Maday and Turinici [J. Chem. Phys. (2003) 8191–8196]. InMaday et al. [Num. Math. (2006) 323–338], a time discretization which preserves theproperty of monotonicity has been presented. This paper introduces aproof of the convergence of these schemes and some results regarding theirrate of convergence.

We consider the coupling between three-dimensional(3D) and one-dimensional (1D) fluid-structure interaction (FSI) models describing blood flow inside compliant vessels. The 1D model is a hyperbolicsystem of partial differential equations.The 3D model consists of the Navier-Stokes equationsfor incompressible Newtonian fluids coupled with a model for the vessel wall dynamics. A non standard formulationfor the Navier-Stokes equations is adopted tohave suitable boundary conditions for the couplingof the models. With this we derive an energy estimatefor the fully 3D-1D FSI coupling. We consider several possiblemodels for the mechanics of the vessel wall in the 3D problemand show how the 3D-1D coupling depends on them.Several comparative numerical tests illustrating the coupling are presented.