In this paper we investigate the motion of a rigid ball in an incompressible perfect fluid occupying ${\mathbb R}^2$. We prove the global in time existence and the uniqueness of the classical solution for this fluid-structure problem. The proof relies mainly on weighted estimates for the vorticity associated with the strong solution of a fluid-structure problem obtained by incorporating some dissipation.

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 review the optimal design of an arterial bypass graft following either a (i) boundaryoptimal control approach, or a (ii) shape optimization formulation. The main focus isquantifying and treating the uncertainty in the residual flow when the hosting artery isnot completely occluded, for which the worst-case in terms of recirculation effects isinferred to correspond to a strong orifice flow through near-complete occlusion.Aworst-case optimal control approach is applied to the steady Navier-Stokes equations in 2Dto identify an anastomosis angle and a cuffed shape that are robust with respect to apossible range of residual flows. We also consider a reduced order modelling frameworkbased on reduced basis methods in order to make the robust design problem computationallyfeasible. The results obtained in 2D are compared with simulations in a 3D geometry butwithout model reduction or the robust framework.

In this work we introduce a new class of lowest order methods for diffusive problems on general meshes with only one unknown per element. The underlying idea is to construct an incomplete piecewise affine polynomial space with optimal approximation properties starting from values at cell centers. To do so we borrow ideas from multi-point finite volume methods, although we use them in a rather different context. The incomplete polynomial space replaces classical complete polynomial spaces in discrete formulations inspired by discontinuous Galerkin methods. Two problems are studied in this work: a heterogeneous anisotropic diffusion problem, which is used to lay the pillars of the method, and the incompressible Navier-Stokes equations, which provide a more realistic application. An exhaustive theoretical study as well as a set of numerical examples featuring different difficulties are provided.

Nowadays, the Coupled Cluster (CC) method is the probably most widely used high precisionmethod for the solution of the main equation of electronic structure calculation, thestationary electronic Schrödinger equation. Traditionally, theequations of CC are formulated as a nonlinear approximation of a Galerkin solution of theelectronic Schrödinger equation, i.e. within a given discrete subspace.Unfortunately, this concept prohibits the direct application of concepts of nonlinearnumerical analysis to obtain e.g. existence and uniqueness results orestimates on the convergence of discrete solutions to the full solution. Here, thisshortcoming is approached by showing that based on the choice of anN-dimensional reference subspace R of H1(ℝ3 ×{± 1/2}), the original, continuous electronic Schrödingerequation can be reformulated equivalently as a root equation for an infinite-dimensionalnonlinear Coupled Cluster operator. The canonical projected CC equations may then beunderstood as discretizations of this operator. As the main step, continuity properties ofthe cluster operator S and its adjoint S† asmappings on the antisymmetric energy space H1 are established.

Bidomain models are commonly used for studying and simulatingelectrophysiological waves in the cardiac tissue. Most of thetime, the associated PDEs are solved using explicit finitedifference methods on structured grids. We propose an implicitfinite element method using unstructured grids for an anisotropicbidomain model. The impact and numerical requirements ofunstructured grid methods is investigated using a test casewith re-entrant waves.

We consider the solution of second order elliptic PDEs in Rdwith inhomogeneous Dirichlet data by means of an h–adaptive FEM withfixed polynomial order p ∈ N. As model example serves the Poissonequation with mixed Dirichlet–Neumann boundary conditions, where the inhomogeneousDirichlet data are discretized by use of an H1 / 2–stableprojection, for instance, the L2–projection forp = 1 or the Scott–Zhang projection for general p ≥ 1.For error estimation, we use a residual error estimator which includes the Dirichlet dataoscillations. We prove that each H1 / 2–stable projectionyields convergence of the adaptive algorithm even with quasi–optimal convergence rate.Numerical experiments with the Scott–Zhang projection conclude the work.

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.

This paper presents a model based on spectral hyperviscosity for the simulation of 3D turbulent incompressible flows. One particularity of this model is that the hyperviscosity is active only at the short velocity scales, a feature which is reminiscent of Large Eddy Simulation models. We propose a Fourier–Galerkin approximation of the perturbedNavier–Stokes equations and we show that, as the cutoff wavenumbergoes to infinity, the solution of the modelconverges (up to subsequences) to a weak solution which is dissipativein the sense defined by Duchon and Robert (2000).

We propose a variational analysis for a Black and Scholes equation with stochastic volatility. This equation gives the price of a European option as a function of the time, of the price of the underlying asset and of the volatility when the volatility is a function of a mean reverting Orstein-Uhlenbeck process, possibly correlated with the underlying asset. The variational analysis involves weighted Sobolev spaces. It enables to prove qualitative properties of the solution, namely a maximum principle and additional regularity properties. Finally, we make numerical simulations of the solution, by finite element and finite difference methods.

In this paper we deal with a semilinear hyperbolic chemotaxis model in one space dimension evolving on a network, with suitable transmission conditions at nodes. This framework is motivated by tissue-engineering scaffolds used for improving wound healing. We introduce a numerical scheme, which guarantees global mass densities conservation. Moreover our scheme is able to yield a correct approximation of the effects of the source term at equilibrium. Several numerical tests are presented to show the behavior of solutions and to discuss the stability and the accuracy of our approximation.

In this work, the error behaviour of high-order exponential operator splitting methods for the time integration of nonlinear evolutionary Schrödinger equations is investigated. The theoretical analysis utilises the framework of abstract evolution equations on Banach spaces and the formal calculus of Lie derivatives. The general approach is substantiated on the basis of a convergence result for exponential operator splitting methods of (nonstiff) order p applied to the multi-configuration time-dependent Hartree–Fock (MCTDHF) equations, which are associated with a model reduction for high-dimensional linear Schrödinger equations describing free electrons that interact by Coulomb force. Provided that the analytical solution of the MCTDHF equations constituting a system of coupled linear ordinary differential equations and low-dimensional nonlinear partial differential equations satisfies suitable regularity requirements, convergence of order p − 1 in the H1 Sobolev norm and convergence of order p in the L2 norm is proven. An analogous result follows for the cubic nonlinear Schrödinger equation, which is also illustrated by a numerical experiment.

Discrete-velocity approximations represent a popular way for computing the Boltzmanncollision operator. The direct numerical evaluation of such methods involve a prohibitivecost, typically O(N2d + 1)where d is the dimension of the velocity space. In this paper, followingthe ideas introduced in [C. Mouhot and L. Pareschi, C. R. Acad. Sci. Paris Sér. IMath. 339 (2004) 71–76, C. Mouhot and L. Pareschi, Math.Comput. 75 (2006) 1833–1852], we derive fast summation techniquesfor the evaluation of discrete-velocity schemes which permits to reduce the computationalcost from O(N2d + 1) to O(N̅dNd log2N), N̅ ≪ N, with almost no loss of accuracy.

This paper presents the numerical analysis for a variational formulation of rate-independent phase transformations in elastic solids due to Mielke et al. The new model itself suggests an implicit time-discretization which is combined with thefinite element method in space. A priori error estimates are established for the quasioptimal spatial approximation of the stress fieldwithin one time-step. A posteriorierror estimates motivate an adaptive mesh-refining algorithm for efficient discretization. The proposed scheme enables numerical simulations which show that the model allows for hysteresis.

In this paper, we propose a method for the approximation of the solution ofhigh-dimensional weakly coercive problems formulated in tensor spaces using low-rankapproximation formats. The method can be seen as a perturbation of a minimal residualmethod with a measure of the residual corresponding to the error in a specified solutionnorm. The residual norm can be designed such that the resulting low-rank approximationsare optimal with respect to particular norms of interest, thus allowing to take intoaccount a particular objective in the definition of reduced order approximations ofhigh-dimensional problems. We introduce and analyze an iterative algorithm that is able toprovide an approximation of the optimal approximation of the solution in a given low-ranksubset, without any a priori information on this solution. We alsointroduce a weak greedy algorithm which uses this perturbed minimal residual method forthe computation of successive greedy corrections in small tensor subsets. We prove itsconvergence under some conditions on the parameters of the algorithm. The proposednumerical method is applied to the solution of a stochastic partial differential equationwhich is discretized using standard Galerkin methods in tensor product spaces.

We present an efficient approach for reducing the statistical uncertaintyassociated with direct Monte Carlo simulations of the Boltzmann equation.As with previous variance-reduction approaches, the resulting relativestatistical uncertainty in hydrodynamic quantities (statistical uncertainty normalized by thecharacteristic value of quantity of interest) is smalland independent of the magnitude of the deviation from equilibrium,making the simulation of arbitrarily small deviations from equilibriumpossible. In contrast to previous variance-reduction methods, themethod presented here is able to substantially reduce variance withvery little modification to the standard DSMC algorithm. This is achievedby introducing an auxiliary equilibrium simulation which, via an importanceweight formulation, uses the same particle data as the non-equilibrium(DSMC) calculation; subtracting the equilibrium from the non-equilibriumhydrodynamic fields drastically reduces the statistical uncertaintyof the latter because the two fields are correlated. The resulting formulation is simple to code and provides considerable computational savings for a wide range of problems of practical interest. It is validated by comparing our results with DSMC solutions for steadyand unsteady, isothermal and non-isothermal problems; in all casesvery good agreement between the two methods is found.

We propose a two point subdivision scheme with parameters to draw curves that satisfy Hermite conditions at both ends of [a,b]. We build three functions f,p and s on dyadic numbers and, using infinite products of matrices, we prove that, under assumptions on the parameters, these functions can be extended by continuity on [a,b], with f'=p and f''=s .

We consider an uncoupled, modular regularization algorithm for approximation of theNavier-Stokes equations. The method is: Step 1.1: Advance the NSEone time step, Step 1.1: Regularize to obtain the approximation atthe new time level. Previous analysis of this approach has been for specific time steppingmethods in Step 1.1 and simple stabilizations in Step 1.1. In this report we extend the mathematical support for uncoupled,modular stabilization to (i) the more complex and better performing BDF2 timediscretization in Step 1.1, and (ii) more general (linear ornonlinear) regularization operators in Step 1.1. We give a completestability analysis, derive conditions on the Step 1.1regularization operator for which the combination has good stabilization effects,characterize the numerical dissipation induced by Step 1.1, provean asymptotic error estimate incorporating the numerical error of the method used in Step1.1 and the regularizations consistency error in Step 1.1 and provide numerical tests.

We present a phase field approach to wetting problems, related tothe minimization of capillary energy. We discuss in detail boththe Γ-convergence results on which our numerical algorithmare based, and numerical implementation. Two possible choices ofboundary conditions, needed to recover Young's law for the contactangle, are presented. We also consider an extension of theclassical theory of capillarity, in which the introduction of adissipation mechanism can explain and predict the hysteresis ofthe contact angle. We illustrate the performance of the model byreproducing numerically a broad spectrum of experimental results:advancing and receding drops, drops on inclined planes andsuperhydrophobic surfaces.

We consider in this paper the problem of finding appropriate models for Large Eddy Simulations of turbulent incompressible flows from a mathematical point of view. The Smagorinsky model is analyzed and the vorticity formulation of the Navier–Stokes equations is used to explore more efficient subgrid-scale models as minimal regularizations of these equations. Two classes of variants of the Smagorinsky model emerge from this approach: a model based on anisotropic turbulent viscosity and a selective model based on vorticity angles. The efficiency of these models is demonstrated by comparisons with reference results on decaying turbulence experiments.