We propose a model for a medical device, called a stent, designed for the treatment of cerebral aneurysms. The stent consists of a grid, immersed in the blood flow and located at the inlet of the aneurysm. It aims at promoting a clot within the aneurysm. The blood flow is modelled by the incompressible Navier-Stokes equations and the stent by a dissipative surface term. We propose a stabilized finite element method for this model and we analyse its convergence in the case of the Stokes equations. We present numerical results for academical test cases, and on a realistic aneurysm obtained from medical imaging.

We develop the analysis of stabilized sparse tensor-productfinite element methods for high-dimensional,non-self-adjoint and possibly degenerate second-order partialdifferential equations of the form $-a:\nabla\nabla u + b \cdot \nabla u + cu = f(x)$ , $x \in\Omega = (0,1)^d \subset \mathbb{R}^d$ ,where $a \in \mathbb{R}^{d\times d}$ is a symmetric positive semidefinite matrix,using piecewise polynomials ofdegree p ≥ 1. Our convergence analysis is based on newhigh-dimensional approximation results in sparse tensor-productspaces. We show that the error between the analytical solution u and its stabilizedsparse finite element approximation u h on a partition ofΩ of mesh size h = hL = 2-L satisfies thefollowing bound in the streamline-diffusion norm $|||\cdot|||_{\rm SD}$ ,provided u belongs to the space $\mathcal{H}^{k+1}(\Omega)$ of functionswith square-integrable mixed (k+1)st derivatives: \[|||u-u_h|||_{\rm SD}\leq C_{p,t} d^2 \max\{(2-p)_+,\kappa_0^{d-1},\kappa_1^d\} (|\sqrt{a}| h_L^t+ |b|^{\frac{1}{2}} h_L^{t+\frac{1}{2}} + c^{\frac{1}{2}} h_L^{t+1} \!)|u|_{\mathcal{H}^{t+1}(\Omega)}, \qquad \qquad \qquad\] where $\kappa_i=\kappa_i(p,t,L)$ , i=0,1, and $1 \leq t \leq \min(k,p)$ .We show, under various mild conditionsrelating L to p, L to d, or p to d,that in the case of elliptic transport-dominateddiffusion problems $\kappa_0, \kappa_1 \in (0,1)$ , and hence for p ≥ 1 the'error constant' $C_{p,t} d^2 \max\{(2-p)_+,\kappa_0^{d-1},\kappa_1^d\}$ exhibits exponential decay as d → ∞; in the case of ageneral symmetric positive semidefinite matrix a,the error constant is shown to grow no faster than $\mathcal{O}(d^2)$ .In any case, in the absence of assumptions that relate L, p and d,the error $|||u - u_h|||_{\rm SD}$ is still bounded by $\kappa_\ast^{d-1}|\log_2 h_L|^{d-1}\mathcal{O}(|\sqrt{a}| h_L^t+ |b|^{\frac{1}{2}} h_L^{t+\frac{1}{2}}+c^{\frac{1}{2}} h_L^{t+1})$ , where $\kappa_\ast \in (0,1)$ for all L, p, d ≥ 2.

We consider the Euler equations for compressible fluidsin a nozzle whose cross-section is variable and may contain discontinuities.We view these equations as a hyperbolic system in nonconservative formand investigate weak solutions in the sense of Dal Maso, LeFloch and Murat [J. Math. Pures Appl.74 (1995) 483–548].Observing that the entropy equality has a fully conservative form,we derive a minimum entropy principle satisfied by entropy solutions.We then establish the stability of a class of numerical approximations for this system.


We prove stability and derive error estimates for the recently introduced central discontinuous Galerkin method, in the context of linear hyperbolic equations with possibly discontinuous solutions. A comparison between the centraldiscontinuous Galerkin method and the regular discontinuousGalerkin method in this context is also made.Numerical experiments are provided to validate the quantitativeconclusions from the analysis.

We prove an a priori error estimate for the hp-version of the boundaryelement method with hypersingular operators on piecewise plane open orclosed surfaces. The underlying meshes are supposed to be quasi-uniform.The solutions of problems on polyhedral or piecewise plane open surfaces exhibittypical singularities which limit the convergence rate of the boundary element method.On closed surfaces, and for sufficiently smooth given data, the solution isH 1-regular whereas, on open surfaces, edge singularities arestrong enough to prevent the solution from being in H 1.In this paper we cover both cases and, in particular, prove an a priorierror estimate for the h-version with quasi-uniform meshes.For open surfaces we prove a convergence like O(h1/2p-1),h being the mesh size and p denoting the polynomial degree.This result had been conjectured previously.

In this paper we introduce and analyze some non-overlapping multiplicative Schwarz methods for discontinuous Galerkin (DG) approximations of elliptic problems. The construction of the Schwarz preconditioners is presented in a unified framework for a wide class of DG methods.For symmetric DG approximations we provide optimal convergence bounds for the corresponding error propagation operator, and we show that the resulting methods can be accelerated by using suitable Krylov space solvers. A discussion on the issue of preconditioning non-symmetric DG approximations of elliptic problems is also included. Extensive numerical experiments to confirm the theoretical results and to assess the robustness and the efficiency of the proposed preconditioners are provided.

This paper is concerned with numerical methods for compressible multicomponent fluids. The fluid components are assumed immiscible, and areseparated by material interfaces, each endowed with its own equation of state (EOS). Cell averages of computational cells that are occupiedby several fluid components require a “mixed-cell” EOS, which may not always be physically meaningful, and often leads to spurious oscillations. We present a new interface tracking algorithm, which avoids using mixed-cell information by solving the Riemann problembetween its single-fluid neighboring cells. The resulting algorithm is oscillation-free for isolated material interfaces, conservative, andtends to produce almost perfect jumps across material fronts. The computational framework is general and may be used in conjunction withone's favorite finite-volume method. The robustness of the method is illustrated on shock-interface interaction in one space dimension,oscillating bubbles with radial symmetry and shock-bubble interaction in two space dimensions.

We analyze a force-based quasicontinuum approximation to aone-dimensional system of atoms that interact by a classicalatomistic potential. This force-based quasicontinuum approximationcan be derived as the modification of an energy-basedquasicontinuum approximation by the addition of nonconservativeforces to correct nonphysical “ghost” forces that occur in theatomistic to continuum interface during constant strain. The algorithmicsimplicity and consistency with the purely atomistic modelat constant strain has made the force-basedquasicontinuum approximation popular for large-scalequasicontinuum computations.We prove that the force-based quasicontinuum equations havea unique solution when the magnitude of the external forces satisfyexplicit bounds. For Lennard-Jones next-nearest-neighborinteractions, we show that unique solutions existfor external forces that extend the system nearly to its tensile limit.We give an analysis of the convergence of the ghost force iteration methodto solve the equilibrium equations for the force-based quasicontinuum approximation.We show that the ghost force iteration is a contraction and give an analysis for itsconvergence rate.

We consider a body immersed in a perfect gas and moving under the action of a constant force.Body and gas are in thermal equilibrium. We assume a stochastic interaction body/medium: when a particle of the medium hits the body, it is absorbed and immediately re-emitted with a Maxwellian distribution. This system gives rise to a microscopic model of friction. We study the approach of the body velocity V(t) to the limiting velocity $V_\infty$ and prove that, under suitable smallness assumptions, the approach to equilibrium is $$|V(t)-V_\infty|\approx \frac{C}{t^{d+1}},$$ where d is the dimension of the space, and C is a positive constant. This approach is not exponential, as typical in friction problems, and even slower than for the same problem with elastic collisions.

We propose a numerical scheme to compute the motion of a two-dimensional rigid body in a viscous fluid. Our method combines the method of characteristics with a finite element approximation to solve an ALE formulation of the problem. We derive error estimates implying the convergence of the scheme.

We study a depth-averaged model of gravity-driven flows made ofsolid grains and fluid, moving over variable basal surface. In particular, we are interested in applicationsto geophysical flows such as avalanches and debris flows,which typically contain both solid material and interstitial fluid.The model system consists of mass and momentum balance equations for thesolid and fluid components, coupled together by both conservative and non-conservative terms involving the derivatives of the unknowns,and by interphase drag source terms. The system is hyperbolic at leastwhen the difference between solid and fluid velocities is sufficiently small.We solve numerically the one-dimensional model equations by a high-resolution finite volume scheme based on a Roe-type Riemann solver. Well-balancing oftopography source terms is obtained via a technique that includesthese contributions into the wave structure of the Riemann solution. We present and discuss several numerical experiments, including problemsof perturbed steady flows over non-flat bottom surface that show the efficient modeling of disturbances of equilibrium conditions.

In this work we present new numerical methods to simulate the mechanics of head-tape magnetic storage devices. The elastohydrodynamic problem is formulated in terms of a coupled system which is governed by a nonlinear compressible Reynolds equation for the air pressure over the head, and a rod model for the tape displacement. A fixed point algorithm between the solutions of the elastic and hydrodynamic problems is proposed. For the nonlinear Reynolds equation, a characteristics method and a duality algorithm are developed to cope with the convection dominating and nonlinear diffusion features, respectively. Furthermore, in the duality method the convergence and optimal choice of the parameters are analyzed. At each fixed point iteration, in the elastic model a complementarity formulation is required and appropriate numerical techniques are used. For the spatial discretization different finite element spaces are chosen. Finally, numerical test examples illustrate the theoretical results, as well as the good performance in the simulation of real devices.

We study a two-grid scheme fully discrete in time andspace for solving the Navier-Stokes system. In the first step, thefully non-linear problem is discretized in space on a coarse gridwith mesh-size H and time step k. In the second step, theproblem is discretized in space on a fine grid with mesh-size hand the same time step, and linearized around the velocity u H computed in the first step. The two-grid strategy is motivated bythe fact that under suitable assumptions, the contribution ofu H to the error in the non-linear term, is measured in theL 2 norm in space and time, and thus has a higher-order than ifit were measured in the H 1 norm in space. We present thefollowing results: if h = H2 = k, then the global error ofthe two-grid algorithm is of the order of h, the same as wouldhave been obtained if the non-linear problem had been solveddirectly on thefine grid.

This paper is devoted to Eulerian models for incompressible fluid-structure systems. These models are primarily derived for computational purposes as they allow to simulate in a rather straightforward way complex 3D systems. We first analyze the level set model of immersed membranes proposed in [Cottet and Maitre, Math. Models Methods Appl. Sci.16 (2006) 415–438]. We in particular show that this model can be interpreted as a generalization of so-called Korteweg fluids. We then extend this model to more generic fluid-structure systems. In this framework, assuming anisotropy, the membrane model appears as a formal limit system when the elastic body width vanishes. We finally provide some numerical experiments which illustrate this claim.

The model order reduction methodology of reduced basis (RB)techniques offers efficient treatment of parametrized partial differential equations (P2DEs) by providing both approximate solution procedures and efficient error estimates. RB-methods have so far mainly been applied to finite element schemes for elliptic and parabolic problems. In the current study we extend the methodology to general linear evolution schemes such as finite volume schemes for parabolic and hyperbolic evolution equations. The new theoretic contributions are the formulation of a reduced basis approximation scheme for these general evolution problems and the derivation of rigorous a-posteriori error estimates in various norms. Algorithmically, an offline/online decomposition of the scheme and the error estimators is realized in case of affine parameter-dependence of the problem. This is the basis for a rapid online computation in case of multiple simulation requests.We introduce a new offline basis-generation algorithm based on our a-posteriori error estimator which combines ideas from existing approaches. Numerical experiments for an instationary convection-diffusion problem demonstrate the efficient applicability of the approach.

In this work, we consider dynamic frictionless contact with adhesionbetween a viscoelastic body of the Kelvin-Voigt type and astationary rigid obstacle, based on the Signorini's contact conditions.Including the adhesion processes modeled by the bonding field, a newversion of energy function is defined. We use the energy functionto derive a new form of energy balance which is supported by numericalresults. Employing the time-discretization, we establish a numerical formulation and investigate the convergence of numerical trajectories. The fullydiscrete approximation which satisfies the complementarity conditionsis computed by using the nonsmooth Newton's method with the Kanzow-Kleinmichelfunction. Numerical simulations of a viscoelastic beam clamped attwo ends are presented.

The main objective of this paper is to provenew necessary conditions to the existence ofKAM tori. To do so, we develop a set ofexplicit a-priori estimates for smoothsolutions of Hamilton-Jacobi equations,using a combination of methods fromviscosity solutions,KAM and Aubry-Mather theories.These estimatesare validin anyspace dimension, and can be checked numericallyto detect gaps between KAM tori and Aubry-Mather sets.We apply these results to detect non-integrable regions in several examples such as a forced pendulum, two coupled penduli, andthe double pendulum.

We consider a family of quadrilateral or hexahedral mixed hp-finite elements for an incompressible flow problem 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 an arbitrary bound.For multilevel adaptive grids with hanging nodes and a sufficiently small mesh size,we prove the inf-sup condition uniformly with respect to the meshsize and the polynomial degree.

In this work, we consider the quasistatic frictionless contact problem between aviscoelastic piezoelectric body and a deformable obstacle. The linear electro-viscoelasticconstitutive law is employed to model the piezoelectric material and the normal compliancecondition is used to model the contact. The variational formulation is derived in a formof a coupled system for the displacement and electric potential fields. An existence anduniqueness result is recalled. Then, a fully discrete scheme is introduced based on thefinite element method to approximate the spatial variable and an Euler scheme to discretizethe time derivatives. Error estimates are derived on the approximative solutions and,as a consequence, the linear convergence of the algorithm is deduced under suitableregularity conditions. Finally, some two-dimensional examples are presented to demonstratethe performance of the algorithm.

We present in this paper a pressure correction scheme for the barotropic compressible Navier-Stokes equations, which enjoys an unconditional stability property, in the sense that the energy and maximum-principle-based a priori estimates of the continuous problem also hold for the discrete solution.The stability proof is based on two independent results for general finite volume discretizations, both interesting for their own sake: the L 2-stability of the discrete advection operator provided it is consistent, in some sense, with the mass balance and the estimate of the pressure work by means of the time derivative of the elastic potential.The proposed scheme is built in order to match these theoretical results, and combines a fractional-step time discretization of pressure-correction type with a space discretization associating low order non-conforming mixed finite elements and finite volumes.Numerical tests with an exact smooth solution show the convergence of the scheme.