In this paper, we extend the reduced-basis approximations developed earlier for linear elliptic and parabolic partial differential equations with affine parameterdependence to problems involving (a) nonaffine dependence on theparameter, and (b) nonlinear dependence on the field variable.The method replaces the nonaffine and nonlinear terms with a coefficient function approximation which then permits an efficient offline-online computationaldecomposition. We first review the coefficient function approximation procedure: the essential ingredients are (i) a good collateralreduced-basis approximation space, and (ii) a stable and inexpensiveinterpolation procedure. We then apply this approach to linear nonaffine and nonlinear elliptic and parabolic equations; in eachinstance, we discuss the reduced-basis approximation and the associated offline-online computationalprocedures. Numerical results are presented to assess our approach.

We present a family of high-order, essentially non-oscillatory, central schemes for approximating solutions of hyperbolic systems of conservation laws. These schemes are based on a new centered version of the Weighed Essentially Non-Oscillatory (WENO) reconstruction of point-valuesfrom cell-averages, which is then followed by an accurate approximationof the fluxes via a natural continuous extension of Runge-Kutta solvers.We explicitly construct the third and fourth-order scheme and demonstrate theirhigh-resolution properties in several numerical tests.

A number of approaches for discretizing partial differential equations with random dataare based on generalized polynomial chaos expansions of random variables. These constitutegeneralizations of the polynomial chaos expansions introduced by Norbert Wiener toexpansions in polynomials orthogonal with respect to non-Gaussian probability measures. Wepresent conditions on such measures which imply mean-square convergence of generalizedpolynomial chaos expansions to the correct limit and complement these with illustrativeexamples.

The model order reduction methodology of reduced basis (RB)techniques offers efficient treatment of parametrized partial differential equations (P2DEs) by providing both approximate solution procedures and efficient error estimates. RB-methods have so far mainly been applied to finite element schemes for elliptic and parabolic problems. In the current study we extend the methodology to general linear evolution schemes such as finite volume schemes for parabolic and hyperbolic evolution equations. The new theoretic contributions are the formulation of a reduced basis approximation scheme for these general evolution problems and the derivation of rigorous a-posteriori error estimates in various norms. Algorithmically, an offline/online decomposition of the scheme and the error estimators is realized in case of affine parameter-dependence of the problem. This is the basis for a rapid online computation in case of multiple simulation requests.We introduce a new offline basis-generation algorithm based on our a-posteriori error estimator which combines ideas from existing approaches. Numerical experiments for an instationary convection-diffusion problem demonstrate the efficient applicability of the approach.

In this paper, we extend the reduced-basis methods and associated a posteriori error estimators developed earlier for elliptic partial differential equations to parabolic problems with affine parameter dependence. The essential new ingredient is the presence of time in the formulation and solution of the problem – we shall “simply” treat time as an additional, albeit special, parameter. First, we introduce the reduced-basis recipe – Galerkin projection onto a space WN spanned by solutions of the governing partial differential equation at N selected points in parameter-time space – and develop a new greedy adaptive procedure to “optimally” construct the parameter-time sample set. Second, we propose error estimation and adjoint procedures that provide rigorous and sharp bounds for the error in specific outputs of interest: the estimates serve a priori to construct our samples, and a posteriori to confirm fidelity. Third, based on the assumption of affine parameter dependence, we develop offline-online computational procedures: in the offline stage, we generate the reduced-basis space; in the online stage, given a new parameter value, we calculate the reduced-basis output and associated error bound. The operation count for the online stage depends only on N (typically small) and the parametric complexity of the problem; the method is thus ideally suited for repeated, rapid, reliable evaluation of input-output relationships in the many-query or real-time contexts.

We present one- and two-dimensional central-upwind schemesfor approximating solutions of the Saint-Venant system with source terms due to bottom topography. The Saint-Venant system has steady-state solutionsin which nonzero flux gradients are exactly balanced by the source terms. It is a challenging problem to preservethis delicate balance with numerical schemes.Small perturbations of these states are also very difficultto compute. Our approach is based on extending semi-discrete central schemes forsystems of hyperbolic conservation laws to balance laws.Special attention is paid to the discretization of the sourceterm such as to preserve stationary steady-statesolutions. We also prove that the second-order version of our schemes preserves the nonnegativity of the height of the water.This important feature allows one to compute solutions for problemsthat include dry areas.

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).

A Discontinuous Galerkin method is used for to thenumerical solution of the time-domain Maxwell equations onunstructured meshes. The method relies on the choice of local basisfunctions, a centered mean approximation for the surface integralsand a second-order leap-frog scheme for advancing in time. The methodis proved to be stable for cases with either metallic or absorbingboundary conditions, for a large class of basis functions. A discrete analog of the electromagnetic energy is conserved formetallic cavities. Convergence is proved for $\mathbb{P}_k$ Discontinuous elements on tetrahedral meshes, as well as a discretedivergence preservation property. Promising numerical examples withlow-order elements show the potential of the method.

This paper presents a stabilization technique for approximating transport equations. The key idea consists in introducing an artificial diffusion based on a two-level decomposition of the approximation space.The technique is proved to have stability and convergenceproperties that are similar to that of the streamline diffusion method.

We propose a quasi-Newton algorithm for solving fluid-structure interaction problems. The basic idea of the method is to build an approximate tangent operator which is cost effective and which takes into account the so-called added mass effect. Various test cases show that the method allows a significant reduction of the computational effort compared to relaxed fixed point algorithms. We present 2D and 3D fluid-structure simulations performed either with a simple 1D structure model or with shells in large displacements.

In this paper, we propose a new diffuse interface model for the study of three immiscible component incompressible viscous flows. The model is based on the Cahn-Hilliard free energy approach. The originality of our study lies in particular in the choice of the bulk free energy. We show that one must take care of this choice in order for the model to give physically relevant results. More precisely, we give conditions for the model to be well-posed and to satisfy algebraically and dynamically consistency properties with the two-component models. Notice that our model is also able to cope with some total spreading situations.We propose to take into account the hydrodynamics of the mixture by coupling our ternary Cahn-Hilliard system and the Navier-Stokes equation supplemented by capillary force terms accounting for surface tension effects between the components. Finally, we present some numerical results which illustrate our analysis and which confirm that our model has a better behavior than other possible similar models.

The convergence and efficiency of the reduced basis method used for the approximation of the solutions to a class of problems written as a parametrized PDE depends heavily on the choice of the elements that constitute the “reduced basis”. The purpose of this paper is to analyze the a priori convergence for one of the approaches used for the selection of these elements, the greedy algorithm. Under natural hypothesis on the set of all solutions to the problem obtained when the parameter varies, we prove that three greedy algorithms converge; the last algorithm, based on the use of an a posteriori estimator, is the approach actually employed in the calculations.

We investigate time harmonic Maxwell equations in heterogeneous media, where thepermeability μ and the permittivity ε are piecewise constant. Theassociated boundary value problem can be interpreted as a transmission problem. Ina very natural way the interfaces can have edges and corners. We give a detaileddescription of the edge and corner singularities of the electromagnetic fields.

This paper is concerned with the numerical approximations of Cauchy problems for one-dimensional nonconservative hyperbolic systems. The first goal is to introduce a general concept of well-balancing for numerical schemes solving this kind of systems. Once this concept stated, we investigate the well-balance properties of numerical schemes based on the generalized Roe linearizations introduced by [Toumi, J. Comp. Phys.102 (1992) 360–373]. Next, this general theory is applied to obtain well-balanced schemes for solving coupled systems of conservation laws with source terms. Finally, we focus on applications to shallow water systems: the numerical schemes obtained and their properties are compared, in the case of one layer flows, with those introduced by [Bermúdez and Vázquez-Cendón, Comput. Fluids23 (1994) 1049–1071]; in the case of two layer flows, they are compared with the numerical scheme presented by [Castro, Macías and Parés, ESAIM: M2AN35 (2001) 107–127].

We developed a mimetic finite difference method for solving elliptic equations with tensor coefficients on polyhedral meshes. The first-order convergence estimates in a mesh-dependent H1 norm are derived.

In this paper, a class of cell centered finite volume schemes, on general unstructured meshes, for a linear convection-diffusion problem, is studied. The convection and the diffusion are respectively approximated by means of an upwind scheme and the so called diamond cell method [4]. Our main result is an error estimate oforder h, assuming only the W2,p (for p>2) regularity of thecontinuous solution, on a mesh of quadrangles. The proof is based on anextension of the ideas developed in [12]. Some newdifficulties arise here, due to the weak regularity of the solution, and thenecessity to approximate the entire gradient, and not only its normalcomponent, as in [12].

We present a finite volume method based on the integration of the Laplaceequation on both the cells of a primal almost arbitrary two-dimensionalmesh and those of adual mesh obtained by joining the centers of the cells of the primal mesh.The key ingredient is the definition of discrete gradient and divergenceoperators verifying a discrete Green formula.This method generalizes an existing finite volume method thatrequires “Voronoi-type” meshes.We show the equivalence of this finite volume method with a non-conformingfinite element method with basis functions being P 1 on the cells,generally called “diamond-cells”, of a third mesh. Under geometrical conditions on these diamond-cells, we prove a first-order convergence both in the $\xHone_0$ normand in the L² norm. Superconvergence results are obtained on certain types of homothetically refined grids. Finally, numerical experiments confirm these results and also show second-order convergencein the L² norm on general grids.They also indicate that this method performs particularly well for the approximationof the gradient of the solution, and may be used on degenerating triangular grids.An example of application on non-conforming locally refined grids is given.

This study concerns some asymptotic models used to compute the flow outside and inside fractures in a bidimensional porous medium. The flow is governed by the Darcy law both in the fractures and in the porous matrix with large discontinuities in the permeability tensor. These fractures are supposed to have a small thickness with respect to the macroscopic length scale,so that we can asymptotically reduce them to immersed polygonal faultinterfaces and the model finally consists in a coupling between a2D elliptic problem and a 1D equation on the sharp interfaces modelling the fractures.A cell-centered finite volume scheme on general polygonal meshes fitting the interfacesis derived to solve the set of equations with the additional differential transmission conditions linking both pressure and normal velocityjumps through the interfaces.We prove the convergence of the FV scheme for any set of data and parameters of the models and derive existence and uniqueness of the solution to the asymptotic models proposed. The models are then numerically experimented for highly or partially immersed fractures.Some numerical results are reported showing different kinds of flowsin the case of impermeable or partially/highly permeable fractures. The influence of the variation of the aperture of the fractures is also investigated. The numericalsolutions of the asymptotic models are validated by comparing them to the solutions of the global Darcy model or to some analytic solutions.

In this note, we propose and analyse a method for handlinginterfaces between non-matching grids based on an approachsuggested by Nitsche (1971) for the approximation ofDirichlet boundary conditions. The exposition is limited toself-adjoint elliptic problems, using Poisson's equation as amodel. A priori and a posteriori error estimates are given. Somenumerical results are included.

We consider H(curl;Ω)-elliptic problems that have been discretized bymeans of Nédélec's edge elements on tetrahedral meshes. Suchproblems occur in the numerical computation of eddy currents. From the defectequation we derive localized expressions that can be used as a posteriori error estimators to control adaptiverefinement. Under certain assumptions on material parameters and computational domains, we derive local lower bounds and a global upper bound for the total error measured in the energy norm. The fundamental tool in the numerical analysis is a Helmholtz-type decomposition of the error into an irrotational part and a weakly solenoidal part.