The parareal in time algorithm allows for efficient parallel numerical simulations oftime-dependent problems. It is based on a decomposition of the time interval intosubintervals, and on a predictor-corrector strategy, where the propagations over eachsubinterval for the corrector stage are concurrently performed on the different processorsthat are available. In this article, we are concerned with the long time integration ofHamiltonian systems. Geometric, structure-preserving integrators are preferably employedfor such systems because they show interesting numerical properties, in particularexcellent preservation of the total energy of the system. Using a symmetrization procedureand/or a (possibly also symmetric) projection step, we introduce here several variants ofthe original plain parareal in time algorithm [L. Baffico, et al. Phys. Rev. E66 (2002) 057701; G. Bal and Y. Maday, A parareal timediscretization for nonlinear PDE’s with application to the pricing of an American put, inRecent developments in domain decomposition methods, Lect.Notes Comput. Sci. Eng. 23 (2002) 189–202; J.-L. Lions, Y. Madayand G. Turinici, C. R. Acad. Sci. Paris, Série I 332 (2001)661–668.] that are better adapted to the Hamiltonian context. These variants arecompatible with the geometric structure of the exact dynamics, and are easy to implement.Numerical tests on several model systems illustrate the remarkable properties of theproposed parareal integrators over long integration times. Some formal elements ofunderstanding are also provided.

We study preconditioning techniques for discontinuous Galerkin discretizations of isotropic linear elasticity problems in primal (displacement) formulation. We propose subspace correction methods based on a splitting of the vector valued piecewise linear discontinuous finite element space, that are optimal with respect to the mesh size and the Lamé parameters. The pure displacement, the mixed and the traction free problems are discussed in detail. We present a convergence analysis of the proposed preconditioners and include numerical examples that validate the theory and assess the performance of the preconditioners.

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 consider the problem of providing optimal uncertainty quantification (UQ) – and hencerigorous certification – for partially-observed functions. We present a UQ frameworkwithin which the observations may be small or large in number, and need not carryinformation about the probability distribution of the system in operation. The UQobjectives are posed as optimization problems, the solutions of which are optimal boundson the quantities of interest; we consider two typical settings, namely parametersensitivities (McDiarmid diameters) and output deviation (or failure) probabilities. Thesolutions of these optimization problems depend non-trivially (even non-monotonically anddiscontinuously) upon the specified legacy data. Furthermore, the extreme values are oftendetermined by only a few members of the data set; in our principal physically-motivatedexample, the bounds are determined by just 2 out of 32 data points, and the remaindercarry no information and could be neglected without changing the final answer. We proposean analogue of the simplex algorithm from linear programming that uses these observationsto offer efficient and rigorous UQ for high-dimensional systems with high-cardinalitylegacy data. These findings suggest natural methods for selecting optimal (maximallyinformative) next experiments.

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.

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.

Lower and upper bounds for the Rayleigh conductivity of a perforation in a thick plate are usually derived from intuitive approximations and by physical reasoning. This paper addresses a mathematical justification of these approaches. As a byproduct of the rigorous handling of these issues, some improvements to previous bounds for axisymmetric holes are given as well as new estimates for tilted perforations. The main techniques are a proper use of the Dirichlet and Kelvin variational principlesin the context of Beppo-Levi spaces. The derivations are validated by numerical experiments in 2D for the axisymmetric case as well as for the full three-dimensional problem.

The surface Cauchy–Born (SCB) method is a computational multi-scale method for the simulation of surface-dominated crystalline materials. We present an error analysis of the SCB method, focused on the role of surface relaxation. In a linearized 1D model we show that the error committed by the SCB method is 𝒪(1) in the mesh size; however, we are able to identify an alternative “approximation parameter” – the stiffness of the interaction potential – with respect to which the relative error in the mean strain is exponentially small. Our analysis naturally suggests an improvement of the SCB model by enforcing atomistic mesh spacing in the normal direction at the free boundary. In this case we even obtain pointwise error estimates for the strain.

For a two phase incompressible flow we consider a diffuse interface model aimed at addressing the movement of three-phase (fluid-fluid-solid) contact lines. The model consists of the Cahn Hilliard Navier Stokes system with a variant of the Navier slip boundary conditions. We show that this model possesses a natural energy law. For this system, a new numerical technique based on operator splitting and fractional time-stepping is proposed. The method is shown to be unconditionally stable. We present several numerical illustrations of this scheme.

An extension of the local projection stabilization (LPS) finite element method for convection-diffusion-reaction equations is presented and analyzed, both in the steady-state and the transient setting. In addition to the standard LPS method, a nonlinear crosswind diffusion term is introduced that accounts for the reduction of spurious oscillations. The existence of a solution can be proved and, depending on the choice of the stabilization parameter, also its uniqueness. Error estimates are derived which are supported by numerical studies. These studies demonstrate also the reduction of the spurious oscillations.

In the article an optimal control problem subject to a stationary variational inequalityis investigated. The optimal control problem is complemented with pointwise controlconstraints. The convergence of a smoothing scheme is analyzed. There, the variationalinequality is replaced by a semilinear elliptic equation. It is shown that solutions ofthe regularized optimal control problem converge to solutions of the original one. Passingto the limit in the optimality system of the regularized problem allows to proveC-stationarity of local solutions of the original problem. Moreover, convergence rateswith respect to the regularization parameter for the error in the control are obtained,which turn out to be sharp. These rates coincide with rates obtained by numericalexperiments, which are included in the paper.

Taking the cue from stabilized Galerkin methods for scalar advection problems, we adaptthe technique to boundary value problems modeling the advection of magnetic fields. Weprovide rigorous a priori error estimates for both fully discontinuouspiecewise polynomial trial functions and -conforming finite elements.

The reliable and effective assimilation of measurements and numerical simulations inengineering applications involving computational fluid dynamics is an emerging problem assoon as new devices provide more data. In this paper we are mainly driven by hemodynamicsapplications, a field where the progressive increment of measures and numerical toolsmakes this problem particularly up-to-date. We adopt a Bayesian approach to the inclusionof noisy data in the incompressible steady Navier-Stokes equations (NSE). The purpose isthe quantification of uncertainty affecting velocity and flow related variables ofinterest, all treated as random variables. The method consists in the solution of anoptimization problem where the misfit between data and velocity - in a convenient norm -is minimized under the constraint of the NSE. We derive classical point estimators, namelythe maximum a posteriori – MAP – and the maximum likelihood – ML – ones.In addition, we obtain confidence regions for velocity and wall shear stress, a flowrelated variable of medical relevance. Numerical simulations in 2-dimensional andaxisymmetric 3-dimensional domains show the gain yielded by the introduction of a completestatistical knowledge in the assimilation process.

Mathematical models for option pricing often result in partial differential equations.Recent enhancements are models driven by Lévy processes, which lead to a partialdifferential equation with an additional integral term. In the context of modelcalibration, these partial integro differential equations need to be solved quitefrequently. To reduce the computational cost the implementation of a reduced order modelhas shown to be very successful numerically. In this paper we give a priorierror estimates for the use of the proper orthogonal decomposition technique inthe context of option pricing models.

Nous étudions l’effet d’une couche mince rugueuse périodique déposée sur une structuresemi-infinie, dans le contexte Helmholtz bi-dimensionnel. Formellement, nous obtenons desconditions de transmission équivalentes à l’ordre 1, par des techniques de typehomogénéisation. Suivent alors la résolution du problème du milieu effectif éclairé parune onde plane, et le calcul de la fonction de Green effective ; le tout par analyse deFourier. Dans un deuxième temps, nous considérons le problème de diffraction par un objetpénétrable enfoui dans la structure recouverte par la couche rugueuse. Nous le résolvonspar la méthode des éléments finis de frontière, dans le milieu effectif. Des résultatsnumériques sont présentés. Enfin, le modèle effectif est validé dans le cas d’une coucheplate, et l’approximation de Born est utilisée pour tester le code des équationsintégrales.

For the efficient numerical solution of indefinite linear systems arising from curl conforming edge element approximations of the time-harmonic Maxwell equation, we consider local multigrid methods (LMM) on adaptively refined meshes. The edge element discretization is done by the lowest order edge elements of Nédélec’s first family. The LMM features local hybrid Hiptmair smoothers of Jacobi and Gauss–Seidel type which are performed only on basis functions associated with newly created edges/nodal points or those edges/nodal points where the support of the corresponding basis function has changed during the refinement process. The adaptive mesh refinement is based on Dörfler marking for residual-type a posteriori error estimators and the newest vertex bisection strategy. Using the abstract Schwarz theory of multilevel iterative schemes, quasi-optimal convergence of the LMM is shown, i.e., the convergence rates are independent of mesh sizes and mesh levels provided the coarsest mesh is chosen sufficiently fine. The theoretical findings are illustrated by the results of some numerical examples.

We introduce and analyze a fully-mixed finite element method for a fluid-solidinteraction problem in 2D. The model consists of an elastic body which is subject to agiven incident wave that travels in the fluid surrounding it. Actually, the fluid issupposed to occupy an annular region, and hence a Robin boundary condition imitating thebehavior of the scattered field at infinity is imposed on its exterior boundary, which islocated far from the obstacle. The media are governed by the elastodynamic and acousticequations in time-harmonic regime, respectively, and the transmission conditions are givenby the equilibrium of forces and the equality of the corresponding normal displacements.We first apply dual-mixed approaches in both domains, and then employ the governingequations to eliminate the displacement u of the solid and the pressure pof the fluid. In addition, since both transmission conditions become essential, they areenforced weakly by means of two suitable Lagrange multipliers. As a consequence, theCauchy stress tensor and the rotation of the solid, together with the gradient ofp and the traces of u and p on the boundary of thefluid, constitute the unknowns of the coupled problem. Next, we show that suitabledecompositions of the spaces to which the stress and the gradient of pbelong, allow the application of the Babuška–Brezzi theory and the Fredholm alternativefor analyzing the solvability of the resulting continuous formulation. The unknowns of thesolid and the fluid are then approximated by a conforming Galerkin scheme defined in termsof PEERS elements in the solid, Raviart–Thomas of lowest order in the fluid, andcontinuous piecewise linear functions on the boundary. Then, the analysis of the discretemethod relies on a stable decomposition of the corresponding finite element spaces andalso on a classical result on projection methods for Fredholm operators of index zero.Finally, some numerical results illustrating the theory are presented.

In vivo visualization of cardiovascular structures ispossible using medical images. However, one has to realize that the resulting 3Dgeometries correspond to in vivo conditions. This entails an internalstress state to be present in the in vivo measured geometry ofe.g. a blood vessel due to the presence of the blood pressure. In orderto correct for this in vivo stress, this paper presents an inverse methodto restore the original zero-pressure geometry of a structure, and to recover thein vivo stress field of the final, loaded structure. The proposedbackward displacement method is able to solve the inverse problem iteratively using fixedpoint iterations, but can be significantly accelerated by a quasi-Newton technique inwhich a least-squares model is used to approximate the inverse of the Jacobian. The hereproposed backward displacement method allows for a straightforward implementation of thealgorithm in combination with existing structural solvers, even if the structural solveris a black box, as only an update of the coordinates of the mesh needs to beperformed.

In this paper we consider a model shape optimization problem. The state variable solvesan elliptic equation on a domain with one part of the boundary described as the graph of acontrol function. We prove higher regularity of the control and develop a priorierror analysis for the finite element discretization of the shape optimizationproblem under consideration. The derived a priori error estimates areillustrated on two numerical examples.

A mixed finite element method for the Navier–Stokes equations is introduced in which thestress is a primary variable. The variational formulation retains the mathematicalstructure of the Navier–Stokes equations and the classical theory extends naturally tothis setting. Finite element spaces satisfying the associated inf–sup conditions aredeveloped.