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 introduce a new variational method for the numerical homogenization of divergence formelliptic, parabolic and hyperbolic equations with arbitrary rough (L∞)coefficients. Our method does not rely on concepts of ergodicity or scale-separation buton compactness properties of the solution space and a new variational approach tohomogenization. The approximation space is generated by an interpolation basis (overscattered points forming a mesh of resolution H) minimizing the L2 norm of thesource terms; its (pre-)computation involves minimizing 𝒪(H−d)quadratic (cell) problems on (super-)localized sub-domains of size 𝒪(H ln(1/H)).The resulting localized linear systems remain sparse and banded. The resultinginterpolation basis functions are biharmonic for d ≤ 3, and polyharmonic ford ≥ 4, forthe operator −div(a∇·) and can be seen as a generalization ofpolyharmonic splines to differential operators with arbitrary rough coefficients. Theaccuracy of the method (𝒪(H)in energy norm and independent from aspect ratios of the mesh formed by the scatteredpoints) is established via the introduction of a new class ofhigher-order Poincaré inequalities. The method bypasses (pre-)computations on the fulldomain and naturally generalizes to time dependent problems, it also provides a naturalsolution to the inverse problem of recovering the solution of a divergence form ellipticequation from a finite number of point measurements.

In a recent paper [E. Chacón Vera and D. Franco Coronil, J. Numer. Math. 20 (2012) 161–182.] a non standard mortar method for incompressible Stokes problem was introduced where the use of the trace spaces H1 / 2and H1/200and a direct computation of the pairing of the trace spaces with their duals are the main ingredients. The importance of the reduction of the number of degrees of freedom leads naturally to consider the stabilized version and this is the results we present in this work. We prove that the standard Brezzi–Pitkaranta stabilization technique is available and that it works well with this mortar method. Finally, we present some numerical tests to illustrate this behaviour.

We derive asymptotic formulas for the solutions of the mixed boundary value problem forthe Poisson equation on the union of a thin cylindrical plate and several thin cylindricalrods. One of the ends of each rod is set into a hole in the plate and the other one issupplied with the Dirichlet condition. The Neumann conditions are imposed on the wholeremaining part of the boundary. Elements of the junction are assumed to have contrastingproperties so that the small parameter, i.e. the relative thickness,appears in the differential equation, too, while the asymptotic structures cruciallydepend on the contrastness ratio. Asymptotic error estimates are derived in anisotropicweighted Sobolev norms.

This paper develops a framework to include Dirichlet boundary conditions on a subset ofthe boundary which depends on time. In this model, the boundary conditions are weaklyenforced with the help of a Lagrange multiplier method. In order to avoid that the ansatzspace of the Lagrange multiplier depends on time, a bi-Lipschitz transformation, whichmaps a fixed interval onto the Dirichlet boundary, is introduced. An inf-sup condition aswell as existence results are presented for a class of second order initial-boundary valueproblems. For the semi-discretization in space, a finite element scheme is presented whichsatisfies a discrete stability condition. Because of the saddle point structure of theunderlying PDE, the resulting system is a DAE of index 3.

This paper deals with the non-conservative coupling of two one-dimensional barotropic Euler systems at an interface at x = 0. The closure pressure laws differ in the domains x < 0 and x > 0, and a Dirac source term concentrated at x = 0 models singular pressure losses. We propose two numerical methods. The first one relies on ghost state reconstructions at the interface while the second is based on a suitable relaxation framework. Both methods satisfy a well-balanced property for stationary solutions. In addition, the second method preserves mass conservation and exactly restores the prescribed singular pressure drops for both unsteady and steady solutions.

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).

Compatible schemes localize degrees of freedom according to the physical nature of the underlying fields and operate a clear distinction between topological laws and closure relations. For elliptic problems, the cornerstone in the scheme design is the discrete Hodge operator linking gradients to fluxes by means of a dual mesh, while a structure-preserving discretization is employed for the gradient and divergence operators. The discrete Hodge operator is sparse, symmetric positive definite and is assembled cellwise from local operators. We analyze two schemes depending on whether the potential degrees of freedom are attached to the vertices or to the cells of the primal mesh. We derive new functional analysis results on the discrete gradient that are the counterpart of the Sobolev embeddings. Then, we identify the two design properties of the local discrete Hodge operators yielding optimal discrete energy error estimates. Additionally, we show how these operators can be built from local nonconforming gradient reconstructions using a dual barycentric mesh. In this case, we also prove an optimal L2-error estimate for the potential for smooth solutions. Links with existing schemes (finite elements, finite volumes, mimetic finite differences) are discussed. Numerical results are presented on three-dimensional polyhedral meshes.

A new class of history-dependent quasivariational inequalities was recently studied in[M. Sofonea and A. Matei, History-dependent quasivariational inequalities arising incontact mechanics. Eur. J. Appl. Math. 22 (2011) 471–491].Existence, uniqueness and regularity results were proved and used in the study of severalmathematical models which describe the contact between a deformable body and an obstacle.The aim of this paper is to provide numerical analysis of the quasivariationalinequalities introduced in the aforementioned paper. To this end we introduce temporallysemi-discrete and fully discrete schemes for the numerical approximation of theinequalities, show their unique solvability, and derive error estimates. We then applythese results to a quasistatic frictional contact problem in which the material’s behavioris modeled with a viscoelastic constitutive law, the contact is bilateral, and friction isdescribed with a slip-rate version of Coulomb’s law. We discuss implementation of thecorresponding fully-discrete scheme and present numerical simulation results on atwo-dimensional example.

We study the time-harmonic acoustic scattering in a duct in presence of a flow and of adiscontinuous impedance boundary condition. Unlike a continuous impedance, a discontinuousone leads to still open modeling questions, as in particular the singularity of thesolution at the abrupt transition and the choice of the right unknown to formulate thescattering problem. To address these questions we propose a mathematical approach based onvariational formulations set in weighted Sobolev spaces. Considering the discontinuousimpedance as the limit of a continuous boundary condition, we prove that only the problemformulated in terms of the velocity potential converges to a well-posed problem. Moreoverwe identify the limit problem and determine some Kutta-like condition satisfied by thevelocity: its convective derivative must vanish at the ends of the impedance area. Finallywe justify why it is not possible to define limit problems for the pressure and thedisplacement. Numerical examples illustrate the convergence process.

We discuss new MUSCL reconstructions to approximate the solutions of hyperbolic systems of conservations laws on 2D unstructured meshes. To address such an issue, we write two MUSCL schemes on two overlapping meshes. A gradient reconstruction procedure is next defined by involving both approximations coming from each MUSCL scheme. This process increases the number of numerical unknowns, but it allows to reconstruct very accurate gradients. Moreover a particular attention is paid on the limitation procedure to enforce the required robustness property. Indeed, the invariant region is usually preserved at the expense of a more restrictive CFL condition. Here, we try to optimize this condition in order to reduce the computational cost.

In this paper, we consider a backward problem for a time-fractional diffusion equation with variable coefficients in a general bounded domain. That is to determine the initial data from a noisy final data. Based on a series expression of the solution, a conditional stability for the initial data is given. Further, we propose a modified quasi-boundary value regularization method to deal with the backward problem and obtain two kinds of convergence rates by using an a priori regularization parameter choice rule and an a posteriori regularization parameter choice rule. Numerical examples in one-dimensional and two-dimensional cases are provided to show the effectiveness of the proposed methods.