In the present work we investigate the numerical simulation of liquid-vapor phase changein compressible flows. Each phase is modeled as a compressible fluid equipped with its ownequation of state (EOS). We suppose that inter-phase equilibrium processes in the mediumoperate at a short time-scale compared to the other physical phenomena such as convectionor thermal diffusion. This assumption provides an implicit definition of an equilibriumEOS for the two-phase medium. Within this framework, mass transfer is the result of localand instantaneous equilibria between both phases. The overall model is strictlyhyperbolic. We examine properties of the equilibrium EOS and we propose a discretizationstrategy based on a finite-volume relaxation method. This method allows to cope with theimplicit definition of the equilibrium EOS, even when the model involves complex EOS’s forthe pure phases. We present two-dimensional numerical simulations that shows that themodel is able to reproduce mechanism such as phase disappearance and nucleation.

We introduce a family of mixed discontinuous Galerkin (DG) finite element methods fornearly and perfectly incompressible linear elasticity. These mixed methods allow thechoice of polynomials of any order k ≥ 1 for the approximation of thedisplacement field, and of order k or k − 1 for thepressure space, and are stable for any positive value of the stabilization parameter. Weprove the optimal convergence of the displacement and stress fields in both cases, witherror estimates that are independent of the value of the Poisson’s ratio. These estimatesdemonstrate that these methods are locking-free. To this end, we prove the correspondinginf-sup condition, which for the equal-order case, requires a construction to establishthe surjectivity of the space of discrete divergences on the pressure space. In theparticular case of near incompressibility and equal-order approximation of thedisplacement and pressure fields, the mixed method is equivalent to a displacement methodproposed earlier by Lew et al. [Appel. Math. Res. express3 (2004) 73–106]. The absence of locking of this displacementmethod then follows directly from that of the mixed method, including the uniform errorestimate for the stress with respect to the Poisson’s ratio. We showcase the performanceof these methods through numerical examples, which show that locking may appear ifDirichlet boundary conditions are imposed strongly rather than weakly, as we do here.

We construct a Galerkin finite element method for the numerical approximation of weaksolutions to a general class of coupled FENE-type finitely extensible nonlinear elasticdumbbell models that arise from the kinetic theory of dilute solutions of polymericliquids with noninteracting polymer chains. The class of models involves the unsteadyincompressible Navier–Stokes equations in a bounded domainΩ ⊂ ℝd, d = 2 or 3, forthe velocity and the pressure of the fluid, with an elastic extra-stress tensor appearingon the right-hand side in the momentum equation. The extra-stress tensor stems from therandom movement of the polymer chains and is defined through the associated probabilitydensity function that satisfies a Fokker–Planck type parabolic equation, a crucial featureof which is the presence of a centre-of-mass diffusion term. We require no structuralassumptions on the drag term in the Fokker–Planck equation; in particular, the drag termneed not be corotational. We perform a rigorous passage to the limit as first the spatialdiscretization parameter, and then the temporal discretization parameter tend to zero, andshow that a (sub)sequence of these finite element approximations converges to a weaksolution of this coupled Navier–Stokes–Fokker–Planck system. The passage to the limit isperformed under minimal regularity assumptions on the data: a square-integrable anddivergence-free initial velocity datum \hbox{$\absundertilde$}${\mathit{u}}_{\mathrm{0}}{\mathrm{~}}_{}$ for the Navier–Stokes equation and a nonnegative initial probabilitydensity function ψ0 for the Fokker–Planck equation, which hasfinite relative entropy with respect to the Maxwellian M.

In this paper, we construct and analyze finite element methods for the three dimensionalMonge-Ampère equation. We derive methods using the Lagrange finite element space such thatthe resulting discrete linearizations are symmetric and stable. With this in hand, we thenprove the well-posedness of the method, as well as derive quasi-optimal error estimates.We also present some numerical experiments that back up the theoretical findings.

We examine the composition of the L∞ norm with weaklyconvergent sequences of gradient fields associated with the homogenization of second orderdivergence form partial differential equations with measurable coefficients. Here thesequences of coefficients are chosen to model heterogeneous media and are piecewiseconstant and highly oscillatory. We identify local representation formulas that in thefine phase limit provide upper bounds on the limit superior of theL∞ norms of gradient fields. The local representationformulas are expressed in terms of the weak limit of the gradient fields and localcorrector problems. The upper bounds may diverge according to the presence of roughinterfaces. We also consider the fine phase limits for layered microstructures and forsufficiently smooth periodic microstructures. For these cases we are able to provideexplicit local formulas for the limit of the L∞ norms of theassociated sequence of gradient fields. Local representation formulas for lower bounds areobtained for fields corresponding to continuously graded periodic microstructures as wellas for general sequences of oscillatory coefficients. The representation formulas areapplied to problems of optimal material design.

The Hartree-Fock equation is widely accepted as the basic model of electronic structure calculation which serves as a canonical starting point for more sophisticated many-particle models. We have studied the s∗-compressibility for Galerkin discretizations of the Hartree-Fock equation in wavelet bases. Our focus is on the compression of Galerkin matrices from nuclear Coulomb potentials and nonlinear terms in the Fock operator which hitherto has not been discussed in the literature. It can be shown that the s∗-compressibility is in accordance with convergence rates obtained from best N-term approximation for solutions of the Hartree-Fock equation. This is a necessary requirement in order to achieve numerical solutions for these equations with optimal complexity using the recently developed adaptive wavelet algorithms of Cohen, Dahmen and DeVore.

We consider the symmetric FEM-BEM coupling for the numerical solution of a (nonlinear)interface problem for the 2D Laplacian. We introduce some new a posteriorierror estimators based on the (h − h/2)-errorestimation strategy. In particular, these include the approximation error for the boundarydata, which allows to work with discrete boundary integral operators only. Using theconcept of estimator reduction, we prove that the proposed adaptive algorithm isconvergent in the sense that it drives the underlying error estimator to zero. Numericalexperiments underline the reliability and efficiency of the considered adaptivemesh-refinement.

We consider a strongly magnetized plasma described by a Vlasov-Poisson system with a large external magnetic field. The finite Larmor radius scaling allows to describe its behaviour at very fine scales. We give a new interpretation of the asymptotic equations obtained by Frénod and Sonnendrücker [SIAM J. Math. Anal. 32 (2001) 1227–1247] when the intensity of the magnetic field goes to infinity. We introduce the so-called polarization drift and show that its contribution is not negligible in the limit, contrary to what is usually said. This is due to the non linear coupling between the Vlasov and Poisson equations.

In this paper, we present a superconvergence result for the mixed finite element approximations of general second order elliptic eigenvalue problems. It is known that a superconvergence result has been given by Durán et al. [Math. Models Methods Appl. Sci. 9 (1999) 1165–1178] and Gardini [ESAIM: M2AN 43 (2009) 853–865] for the lowest order Raviart-Thomas approximation of Laplace eigenvalue problems. In this work, we introduce a new way to derive the superconvergence of general second order elliptic eigenvalue problems by general mixed finite element methods which have the commuting diagram property. Some numerical experiments are given to confirm the theoretical analysis.

This work is concerned with a class of minimum effort problems for partial differential equations, where the control cost is of L∞-type. Since this problem is non-differentiable, a regularized functional is introduced that can be minimized by a superlinearly convergent semi-smooth Newton method. Uniqueness and convergence for the solutions to the regularized problem are addressed, and a continuation strategy based on a model function is proposed. Numerical examples for a convection-diffusion equation illustrate the behavior of minimum effort controls.

We propose a numerical analysis of proper orthogonal decomposition (POD) model reductions in which a priori error estimates are expressed in terms of the projection errors that are controlled in the construction of POD bases. These error estimates are derived for generic parabolic evolution PDEs, including with non-linear Lipschitz right-hand sides, and for wave-like equations. A specific projection continuity norm appears in the estimates and – whereas a general uniform continuity bound seems out of reach – we prove that such a bound holds in a variety of Galerkin bases choices. Furthermore, we directly numerically assess this bound – and the effectiveness of the POD approach altogether – for test problems of the type considered in the numerical analysis, and also for more complex equations. Namely, the numerical assessment includes a parabolic equation with super-linear reaction terms, inspired from the FitzHugh-Nagumo electrophysiology model, and a 3D biomechanical heart model. This shows that the effectiveness established for the simpler models is also achieved in the reduced-order simulation of these highly complex systems.

The present work aims at proposing a rigorous analysis of the mathematical and numerical modelling of ultrasonic piezoelectric sensors. This includes the well-posedness of the final model, the rigorous justification of the underlying approximation and the design and analysis of numerical methods. More precisely, we first justify mathematically the classical quasi-static approximation that reduces the electric unknowns to a scalar electric potential. We next justify the reduction of the computation of this electric potential to the piezoelectric domains only. Particular attention is devoted to the different boundary conditions used to model the emission and reception regimes of the sensor. Finally, an energy preserving finite element/finite difference numerical scheme is developed; its stability is analyzed and numerical results are presented.

The purpose of this paper is to provide a priori error estimates on the approximation of contact conditions in the framework of the eXtended Finite-Element Method (XFEM) for two dimensional elastic bodies. This method allows to perform finite-element computations on cracked domains by using meshes of the non-cracked domain. We consider a stabilized Lagrange multiplier method whose particularity is that no discrete inf-sup condition is needed in the convergence analysis. The contact condition is prescribed on the crack with a discrete multiplier which is the trace on the crack of a finite-element method on the non-cracked domain, avoiding the definition of a specific mesh of the crack. Additionally, we present numerical experiments which confirm the efficiency of the proposed method.

We analyze a two-stage implicit-explicit Runge–Kutta scheme for time discretization of advection-diffusion equations. Space discretization uses continuous, piecewise affine finite elements with interelement gradient jump penalty; discontinuous Galerkin methods can be considered as well. The advective and stabilization operators are treated explicitly, whereas the diffusion operator is treated implicitly. Our analysis hinges on L2-energy estimates on discrete functions in physical space. Our main results are stability and quasi-optimal error estimates for smooth solutions under a standard hyperbolic CFL restriction on the time step, both in the advection-dominated and in the diffusion-dominated regimes. The theory is illustrated by numerical examples.

This paper is concerned with a PDE-constrained optimization problem of induction heating, where the state equations consist of 3D time-dependent heat equations coupled with 3D time-harmonic eddy current equations. The control parameters are given by finite real numbers representing applied alternating voltages which enter the eddy current equations via impressed current. The optimization problem is to find optimal voltages so that, under certain constraints on the voltages and the temperature, a desired temperature can be optimally achieved. As there are finitely many control parameters but the state constraint has to be satisfied in an infinite number of points, the problem belongs to a class of semi-infinite programming problems. We present a rigorous analysis of the optimization problem and a numerical strategy based on our theoretical result.

We consider a nonlinear Dirac system in one space dimension with periodic boundary conditions. First, we discuss questions on the existence and uniqueness of the solution. Then, we propose an implicit-explicit finite difference method for its approximation, proving optimal order a priori error estimates in various discrete norms and showing results from numerical experiments.

In this paper, we develop a multiscale mortar multipoint flux mixed finite element method for second order elliptic problems. The equations in the coarse elements (or subdomains) are discretized on a fine grid scale by a multipoint flux mixed finite element method that reduces to cell-centered finite differences on irregular grids. The subdomain grids do not have to match across the interfaces. Continuity of flux between coarse elements is imposed via a mortar finite element space on a coarse grid scale. With an appropriate choice of polynomial degree of the mortar space, we derive optimal order convergence on the fine scale for both the multiscale pressure and velocity, as well as the coarse scale mortar pressure. Some superconvergence results are also derived. The algebraic system is reduced via a non-overlapping domain decomposition to a coarse scale mortar interface problem that is solved using a multiscale flux basis. Numerical experiments are presented to confirm the theory and illustrate the efficiency and flexibility of the method.

Optimal nonanticipating controls are shown to exist in nonautonomous piecewisedeterministic control problems with hard terminal restrictions. The assumptions needed arecompletely analogous to those needed to obtain optimal controls in deterministic controlproblems. The proof is based on well-known results on existence of deterministic optimalcontrols.

This paper analyzes the continuum model/complete electrode model in the electricalimpedance tomography inverse problem of determining the conductivity parameter fromboundary measurements. The continuity and differentiability of the forward operator withrespect to the conductivity parameter inLp-norms are proved. These analytical resultsare applied to several popular regularization formulations, which incorporate apriori information of smoothness/sparsity on the inhomogeneity through Tikhonovregularization, for both linearized and nonlinear models. Some important properties,e.g., existence, stability, consistency andconvergence rates, are established. This provides some theoretical justifications of theirpractical usage.