Subsurface flows are influenced by the presence of faults and large fractures which actas preferential paths or barriers for the flow. In literature models were proposed tohandle fractures in a porous medium as objects of codimension 1. In this work we considerthe case of a network of intersecting fractures, with the aim of deriving physicallyconsistent and effective interface conditions to impose at the intersection betweenfractures. This new model accounts for the angle between fractures at the intersectionsand allows for jumps of pressure across intersections. This fact permits to describe theflow when fractures are characterized by different properties more accurately with respectto other models that impose pressure continuity. The main mathematical properties of themodel, derived in the two-dimensional setting, are analyzed. As concerns the numericaldiscretization we allow the grids of the fractures to be independent, thus in generalnon-matching at the intersection, by means of the extended finite element method(XFEM). This increases the flexibility of the method in the case of complexgeometries characterized by a high number of fractures.

The aim of this paper is to give a simple, introductory presentation of the extension of the Virtual Element Method to the discretization of H(div)-conforming vector fields (or, more generally, of (n − 1) − Cochains). As we shall see, the methods presented here can be seen as extensions of the so-called BDM family to deal with more general element geometries (such as polygons with an almost arbitrary geometry). For the sake of simplicity, we limit ourselves to the 2-dimensional case, with the aim of making the basic philosophy clear. However, we consider an arbitrary degree of accuracy k (the Virtual Element analogue of dealing with polynomials of arbitrary order in the Finite Element Framework).

The paper is concerned with the finite element solution of the Poisson equation with homogeneous Dirichlet boundary condition in a three-dimensional domain. Anisotropic, graded meshes from a former paper are reused for dealing with the singular behaviour of the solution in the vicinity of the non-smooth parts of the boundary. The discretization error is analyzed for the piecewise linear approximation in the H1(Ω)- and L2(Ω)-norms by using a new quasi-interpolation operator. This new interpolant is introduced in order to prove the estimates for L2(Ω)-data in the differential equation which is not possible for the standard nodal interpolant. These new estimates allow for the extension of certain error estimates for optimal control problems with elliptic partial differential equations and for a simpler proof of the discrete compactness property for edge elements of any order on this kind of finite element meshes.

A sporting league places every team into one of several divisions of equal size, and runs a round robin tournament for each division. Some teams are paired with another team, not necessarily in the same division, to share facilities. It is shown that however many teams are paired and whatever the pairings, it is always possible to schedule the fixtures in the minimum time, so that no two paired teams have home matches simultaneously.

We present the current Reduced Basis framework for the efficient numerical approximation of parametrized steady Navier–Stokes equations. We have extended the existing setting developed in the last decade (see e.g. [S. Deparis, SIAM J. Numer. Anal. 46 (2008) 2039–2067; A. Quarteroni and G. Rozza, Numer. Methods Partial Differ. Equ. 23 (2007) 923–948; K. Veroy and A.T. Patera, Int. J. Numer. Methods Fluids 47 (2005) 773–788]) to more general affine and nonaffine parametrizations (such as volume-based techniques), to a simultaneous velocity-pressure error estimates and to a fully decoupled Offline/Online procedure in order to speedup the solution of the reduced-order problem. This is particularly suitable for real-time and many-query contexts, which are both part of our final goal. Furthermore, we present an efficient numerical implementation for treating nonlinear advection terms in a convenient way. A residual-based a posteriori error estimation with respect to a truth, full-order Finite Element approximation is provided for joint pressure/velocity errors, according to the Brezzi–Rappaz–Raviart stability theory. To do this, we take advantage of an extension of the Successive Constraint Method for the estimation of stability factors and of a suitable fixed-point algorithm for the approximation of Sobolev embedding constants. Finally, we present some numerical test cases, in order to show both the approximation properties and the computational efficiency of the derived framework.

We extend the classical empirical interpolation method [M. Barrault, Y. Maday, N.C.Nguyen and A.T. Patera, An empirical interpolation method: application to efficientreduced-basis discretization of partial differential equations. Compt. Rend. Math.Anal. Num. 339 (2004) 667–672] to a weighted empiricalinterpolation method in order to approximate nonlinear parametric functions with weightedparameters, e.g. random variables obeying various probabilitydistributions. A priori convergence analysis is provided for the proposedmethod and the error bound by Kolmogorov N-width is improved from the recent work [Y.Maday, N.C. Nguyen, A.T. Patera and G.S.H. Pau, A general, multipurpose interpolationprocedure: the magic points. Commun. Pure Appl. Anal. 8(2009) 383–404]. We apply our method to geometric Brownian motion, exponentialKarhunen–Loève expansion and reduced basis approximation of non-affine stochastic ellipticequations. We demonstrate its improved accuracy and efficiency over the empiricalinterpolation method, as well as sparse grid stochastic collocation method.

We deal with an inverse scattering problem whose aim is to determine the thicknessvariation of a dielectric thin coating located on a conducting structure of unknown shape.The inverse scattering problem is solved through the application of the GeneralizedImpedance Boundary Conditions (GIBCs) which contain the thickness, curvature as well asmaterial properties of the coating and they have been obtained in the previous work [B.Aslanyürek, H. Haddar and H.Şahintürk, Wave Motion 48 (2011)681–700] up to the third order with respect to the thickness. After proving uniquenessresults for the inverse problem, the required total field as well as its higher orderderivatives appearing in the GIBCs are obtained by the analytical continuation of themeasured data to the coating surface through the single layer potential representation.The resulting system of non-linear differential equations for the unknown coatingthickness is solved iteratively via the Newton−Raphson method after expanding thethickness function in a series of exponentials. Through the simulations it has been shownthat the approach is effective under the validity conditions of the GIBCs.

Two-phase fluid flows on substrates (i.e. wetting phenomena) are important in many industrial processes, such as micro-fluidics and coating flows. These flows include additional physical effects that occur near moving (three-phase) contact lines. We present a new 2-D variational (saddle-point) formulation of a Stokesian fluid with surface tension that interacts with a rigid substrate. The model is derived by an Onsager type principle using shape differential calculus (at the sharp-interface, front-tracking level) and allows for moving contact lines and contact angle hysteresis and pinning through a variational inequality. Moreover, the formulation can be extended to include non-linear contact line motion models. We prove the well-posedness of the time semi-discrete system and fully discrete method using appropriate choices of finite element spaces. A formal energy law is derived for the semi-discrete and fully discrete formulations and preliminary error estimates are also given. Simulation results are presented for a droplet in multiple configurations to illustrate the method.

We introduce a new stable MINI-element pair for incompressible Stokes equations onquadrilateral meshes, which uses the smallest number of bubbles for the velocity. Thepressure is discretized with the P1-midpoint-edge-continuous elements and each component of the velocity field is done withthe standard Q1-conforming elements enriched byone bubble a quadrilateral. The superconvergence in the pressure of the proposed pair isanalyzed on uniform rectangular meshes, and tested numerically on uniform and non-uniformmeshes.

In this paper we propose a time discretization of a system of two parabolic equationsdescribing diffusion-driven atom rearrangement in crystalline matter. The equationsexpress the balances of microforces and microenergy; the two phase fields are the orderparameter and the chemical potential. The initial and boundary-value problem for theevolutionary system is known to be well posed. Convergence of the discrete scheme to thesolution of the continuous problem is proved by a careful development of uniformestimates, by weak compactness and a suitable treatment of nonlinearities. Moreover, forthe difference of discrete and continuous solutions we prove an error estimate of orderone with respect to the time step.

This paper is devoted to the definition, analysis and implementation of semi-Lagrangian methods as they result from particle methods combined with remeshing. We give a complete consistency analysis of these methods, based on the regularity and momentum properties of the remeshing kernels, and a stability analysis of a large class of second and fourth order methods. This analysis is supplemented by numerical illustrations. We also describe a general approach to implement these methods in the context of hybrid computing and investigate their performance on GPU processors as a function of their order of accuracy.

In this paper, we consider the well-known Fattorini’s criterion for approximate controllability of infinite dimensional linear systems of type y′ = Ay + Bu. We precise the result proved by Fattorini in [H.O. Fattorini, SIAM J. Control 4 (1966) 686–694.] for bounded input B, in the case where B can be unbounded or in the case of finite-dimensional controls. More precisely, we prove that if Fattorini’s criterion is satisfied and if the set of geometric multiplicities of A is bounded then approximate controllability can be achieved with finite dimensional controls. An important consequence of this result consists in using the Fattorini’s criterion to obtain the feedback stabilizability of linear and nonlinear parabolic systems with feedback controls in a finite dimensional space. In particular, for systems described by partial differential equations, such a criterion reduces to a unique continuation theorem for a stationary system. We illustrate such a method by tackling some coupled Navier−Stokes type equations (MHD system and micropolar fluid system) and we sketch a systematic procedure relying on Fattorini’s criterion for checking stabilizability of such nonlinear systems. In that case, the unique continuation theorems rely on local Carleman inequalities for stationary Stokes type systems.

This paper establishes the equivalence between systems described by a single first-order hyperbolic partial differential equation and systems described by integral delay equations. System-theoretic results are provided for both classes of systems (among them converse Lyapunov results). The proposed framework can allow the study of discontinuous solutions for nonlinear systems described by a single first-order hyperbolic partial differential equation under the effect of measurable inputs acting on the boundary and/or on the differential equation. Illustrative examples show that the conversion of a system described by a single first-order hyperbolic partial differential equation to an integral delay system can simplify considerably the stability analysis and the solution of robust feedback stabilization problems.

The Euler−Poinsot rigid bodymotion is a standard mechanical system and it is a model for left-invariant Riemannianmetrics on SO(3). In this article using theSerret−Andoyer variables weparameterize the solutions and compute the Jacobi fields in relation with the conjugatelocus evaluation. Moreover, the metric can be restricted to a 2D-surface, and theconjugate points of this metric are evaluated using recent works on surfaces ofrevolution. Another related 2D-metric on S2 associated to the dynamics of spin particles withIsing coupling is analysed using both geometric techniques and numerical simulations.

Nanopore science, the study of individual nanoscale pores within thin membranes, is a fast-growing field which presents numerous interesting problems for physicists and applied mathematicians. Nanopores are most commonly applied to resistive pulse sensing (RPS) of individual particles suspended in aqueous electrolyte. The form of a resistive pulse is dependent on an array of experimental variables, including electrolyte characteristics, electrophoretic and convective transport, and (especially) pore and particle geometry. The level of analysis required depends on the application, but any broadly useful approach should be simple and flexible, due to the requirement for high data throughput and variations between different experimental systems and specimens. Here we review analytic methods for interpreting RPS experiments for particles in the approximate range 100 nm to 1 $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mu $m, focusing on calculation of resistance change as a function of the particle’s position. We detail a recently developed semi-analytical model and compare the modelled electric field with finite element results. The model can also be used to calculate particle motion, so that the experimental current–time history can be reconstructed. This approach is useful for a wide range of pore and particle geometries, and includes consideration of entrance effects. Tunable elastomeric pores with truncated linear cone geometry are used as a model system.

In models of fluid outflows from point or line sources, an interface is present, and it is forced outwards as time progresses. Although various types of fluid instabilities are possible at the interface, it is nevertheless of interest to know the development of its overall shape with time. If the fluids on either side are of nearly equal densities, it is possible to derive a single nonlinear partial differential equation that describes the interfacial shape with time. Although nonlinear, this equation admits a simple transformation that renders it linear, so that closed-form solutions are possible. Two such solutions are illustrated; for a line source in a planar straining flow and a point source in an axisymmetric background flow. Possible applications in astrophysics are discussed.

In this paper, we study the boundary penalty method for optimal control of unsteadyNavier–Stokes type system that has been proposed as an alternative for Dirichlet boundarycontrol. Existence and uniqueness of solutions are demonstrated and existence of optimalcontrol for a class of optimal control problems is established. The asymptotic behavior ofsolution, with respect to the penalty parameter ϵ, is studied. In particular, we prove convergenceof solutions of penalized control problem to the corresponding solutions of the Dirichletcontrol problem, as the penalty parameter goes to zero. We also derive an optimalitysystem and determine optimal solutions. In order to illustrate the theoretical results andthe practical utility of control, we numerically address the problem of controllingunsteady convection with Soret effect using a gradient-based method. Numerical resultsshow the effectiveness of the approach.

We propose and analyse an alternate approach to a priori error estimates for the semidiscrete Galerkin approximation to a time-dependent parabolic integro-differential equation with nonsmooth initial data. The method is based on energy arguments combined with repeated use of time integration, but without using parabolic-type duality techniques. An optimal $L^2$-error estimate is derived for the semidiscrete approximation when the initial data is in $L^2$. A superconvergence result is obtained and then used to prove a maximum norm estimate for parabolic integro-differential equations defined on a two-dimensional bounded domain.

In this paper we are concerned with a distributed optimal control problem governed by anelliptic partial differential equation. State constraints of box type are considered. Weshow that the Lagrange multiplier associated with the state constraints, which is known tobe a measure, is indeed more regular under quite general assumptions. We discretize theproblem by continuous piecewise linear finite elements and we are able to prove that, forthe case of a linear equation, the order of convergence for the error in L2(Ω) of the controlvariable is h |log h | in dimensions 2 and 3.

We propose a new nonrigid registration algorithm which is based on the optimal control approach. In our previously proposed methods, the Jacobian determinant and the curl vector were used as control functions. In this algorithm, we use a new set of control functions. A main advantage of using the new controls is that the positivity and normalization of the Jacobian determinant are satisfied automatically. Numerical results on large deformation brain images are provided to show the accuracy and efficiency of the algorithm.