We consider the initial value problem for degenerate viscous and inviscid scalar conservation laws where the flux function depends on the spatial location through a"rough"coefficient function k(x). We show that the Engquist-Osher (and hence all monotone)finite difference approximations convergeto the unique entropy solution of the governing equation if, among other demands, k' is in BV, thereby providing alternative (new) existence proofs for entropy solutions of degenerate convection-diffusion equations as well as new convergence results for their finite difference approximations. In the inviscid case, we also provide a rate of convergence. Our convergence proofs are based on deriving a series of a priori estimates and using a general Lp compactness criterion.

In this paper we study the derivation of homogeneous hydrostatic equationsstarting from 2D Euler equations, following for instance[2,9]. We give a convergence result for convex profiles anda divergence result for a particular inflexion profile.

In order to handle the flow of a viscous incompressible fluid in a porous medium with cracks, the thickness of which cannot be neglected, we consider a model which couples the Darcy equations in the medium with the Stokes equations in the cracks by a new boundary condition at the interface, namely the continuity of the pressure. We prove that this model admits a unique solution and propose a mixed formulation of it. Relying on this formulation, we describe a finite element discretization and derive a priori and a posteriori error estimates. We present some numerical experiments that are in good agreement with the analysis.

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.

The goal of our paper is to introduce basis functions for the finite element discretization of a second order linear elliptic operator with rough or highly oscillating coefficients.The proposed basis functions are inspired by the classic idea of componentmode synthesis and exploit an orthogonal decompositionof the trial subspace to minimize the energy. Numerical experiments illustrate the effectiveness of the proposed basis functions.

In the present work we introduce a new family of cell-centered Finite Volume schemes for anisotropic and heterogeneous diffusion operators inspired by the MPFA L method.A very general framework for the convergence study of finite volume methods is provided and then used to establish the convergence of the new method.Fairly general meshes are covered and a computable sufficient criterion for coercivity is provided. In order to guarantee consistency in the presence of heterogeneous diffusivity, we introduce a non-standard test space in $H_0^1$ (Ω) and prove its density. Thorough assessment on a set of anisotropic heterogeneous problems as well as a comparison with classical multi-point Finite Volume methods is provided.

The motion of an incompressible fluid confined to a shallow basin witha slightly varying bottom topography is considered. Coriolis force,surface wind and pressure stresses, together with bottom andlateral friction stresses are taken into account. We introduceappropriate scalings into a three-dimensional anisotropic eddyviscosity model; after averaging on the vertical direction andconsidering some asymptotic assumptions, we obtain a two-dimensionalmodel, which approximates the three-dimensional model at the secondorder with respect to the ratio between the vertical scale and thelongitudinal scale. The derived model is shown to be symmetrizablethrough a suitable change of variables. Finally, we propose somenumerical tests with the aim to validate the proposed model.

We consider discontinuous as well as continuous Galerkin methods for the time discretization of a class of nonlinear parabolic equations. We show existence and local uniqueness and derive optimal order optimal regularity a priori error estimates. We establish the results in an abstract Hilbert space setting and apply them to a quasilinear parabolic equation.

The atomistic to continuum interface for quasicontinuum energiesexhibits nonzero forces under uniform strain that have beencalled ghost forces.In this paper,we prove for a linearization of a one-dimensional quasicontinuum energyaround a uniform strainthat the effect of the ghost forces on the displacementnearly cancels and has a small effect on the error away from the interface.We give optimal order error estimatesthat show that the quasicontinuum displacementconverges to the atomistic displacement at the rate O(h)in the discrete $\ell^\infty$ andw 1,1 norms where h is the interatomic spacing.We also give a proof that the error in the displacement gradientdecays away from the interface to O(h) at distance O(h|logh|)in the atomistic region and distance O(h) in the continuum region.Our work gives an explicit and simplified form for the decay of the effect of theatomistic to continuum coupling error in terms of a general underlying interatomic potential and gives the estimates described above in the discrete $\ell^\infty$ and w1,p norms.

We present in this article two components: these components can in fact serve various goalsindependently, though we consider them here as an ensemble. The first component is a technique forthe rapid and reliable evaluation prediction of linear functional outputs of elliptic (andparabolic) partial differential equations with affine parameter dependence. The essential features are (i) (provably) rapidly convergent globalreduced–basis approximations — Galerkin projection onto a spaceW N spanned by solutions of the governing partial differentialequation at N selected points in parameter space; (ii) a posteriori error estimation — relaxations of the error–residualequation that provide inexpensive yet sharp and rigorous bounds forthe error in the outputs of interest; and (iii) off–line/on–linecomputational procedures — methods which decouple the generationand projection stages of the approximation process. This component is ideally suited — consideringthe operation count of the online stage — for the repeated and rapid evaluation required in thecontext of parameter estimation, design, optimization, andreal–time control. The second component is a framework for distributed simulations. This frameworkcomprises a library providing the necessary abstractions/concepts for distributed simulations and asmall set of tools — namely SimTeXand SimLaB— allowing an easy manipulation of thosesimulations. While the library is the backbone of the framework and is therefore general, thevarious interfaces answer specific needs. We shall describe both components and present how theyinteract.

We present a new semi-discrete central scheme for one-dimensional shallow water flows along channels with non-uniform rectangular cross sections and bottom topography. The scheme preserves the positivity of the water height, and it is preserves steady-states of rest (i.e., it is well-balanced). Along with a detailed description of the scheme, numerous numerical examples are presented for unsteady and steady flows. Comparison with exact solutions illustrate the accuracy and robustness of the numerical algorithm.

In this paper we solve the time-dependent incompressible Navier-Stokesequations by splitting the non-linearity and incompressibility, andusing discontinuous or continuous finite element methods in space. Weprove optimal error estimates for the velocity and suboptimalestimates for the pressure. We present some numerical experiments.

We propose a numerical analysis of proper orthogonal decomposition (POD) model reductions in which a priori error estimates are expressed in terms of the projection errors that are controlled in the construction of POD bases. These error estimates are derived for generic parabolic evolution PDEs, including with non-linear Lipschitz right-hand sides, and for wave-like equations. A specific projection continuity norm appears in the estimates and – whereas a general uniform continuity bound seems out of reach – we prove that such a bound holds in a variety of Galerkin bases choices. Furthermore, we directly numerically assess this bound – and the effectiveness of the POD approach altogether – for test problems of the type considered in the numerical analysis, and also for more complex equations. Namely, the numerical assessment includes a parabolic equation with super-linear reaction terms, inspired from the FitzHugh-Nagumo electrophysiology model, and a 3D biomechanical heart model. This shows that the effectiveness established for the simpler models is also achieved in the reduced-order simulation of these highly complex systems.

We propose a model for a medical device, called a stent, designed for the treatment of cerebral aneurysms. The stent consists of a grid, immersed in the blood flow and located at the inlet of the aneurysm. It aims at promoting a clot within the aneurysm. The blood flow is modelled by the incompressible Navier-Stokes equations and the stent by a dissipative surface term. We propose a stabilized finite element method for this model and we analyse its convergence in the case of the Stokes equations. We present numerical results for academical test cases, and on a realistic aneurysm obtained from medical imaging.

We explain why the conventional argument for deriving the time-dependent Born-Oppenheimer approximation is incomplete and review recent mathematical results, which clarify the situation and at the same time provide a systematic scheme for higher order corrections. We also present a new elementary derivation of the correct second-order time-dependent Born-Oppenheimer approximation and discuss as applicationsthe dynamics near a conical intersection of potential surfaces and reactive scattering.

We consider a two dimensional elastic body submitted to unilateral contact conditions, local frictionand adhesion on a part of his boundary. After discretizing the variational formulation with respectto time we use a smoothing technique to approximate the friction term by an auxiliary problem. A shiftingtechnique enables us to obtain the existence of incremental solutions with bounds independent of the regularization parameter. We finally obtain the existence of a quasistatic solution by passing to thelimit with respect to time.

The convergence of a two-scale FEM for elliptic problems in divergence form with coefficients and geometries oscillating at length scale ε << 1 is analyzed. Full elliptic regularity independent of ε is shownwhen the solution is viewed as mapping from the slow into the fast scale.Two-scale FE spaces which are able to resolve the ε scale of thesolution with work independent of ε and withoutanalytical homogenization are introduced. Robustin ε error estimates for the two-scale FE spaces are proved. Numerical experiments confirm thetheoretical analysis.

This paper analyses the implementation of the generalizedfinite differences method for the HJB equation of stochastic control, introduced by two of the authors in [Bonnans and Zidani,SIAM J. Numer. Anal.41 (2003) 1008–1021]. The computation of coefficients needs tosolve at each point of the grid (and for each control)a linear programming problem.We show here that, for two dimensional problems, this linear programming problem can be solved in O(p max) operations, where p max is the size of the stencil. The method is based on a walk on the Stern-Brocot tree,and on the related filling of the set of positive semidefinite matrices of size two.

Object oriented design has proven itself as a powerful tool in the field of scientific computing. Several software packages, libraries and toolkits exist, in particular in the FEM arenathat follow this design methodology providing extensible, reusable,and flexible software while staying competitive to traditionallydesigned point tools in terms of efficiency. However, the common approach to identify classes is to turn data structures and algorithms of traditional implementations into classes such that the level of abstraction is essentially not raised. In this paper we discuss an alternative way to approach the design challenge which we call “concept oriented design”. We apply this design methodology to Petrov-Galerkin methods leading to a class library for both, boundary element methods (BEM) and finite element methods (FEM). We show as a particular example the implementation of hp-FEM using the library with specialattention to the handling of inconsistent meshes.

The LBB condition is well-known to guarantee the stability of a finiteelement (FE) velocity - pressure pair in incompressible flow calculations.To ensure the condition to be satisfied a certain constant should be positive andmesh-independent. The paper studies the dependence of the LBB condition on thedomain geometry. For model domains such as strips and rings thesubstantial dependence of this constant on geometry aspect ratios is observed.In domains with highly anisotropic substructures this may require special care with numerics to avoid failures similar to those whenthe LBB condition is violated. In the core of the paperwe prove that for any FE velocity-pressure pair satisfying usual approximationhypotheses the mesh-independent limit in the LBB condition is not greater thanits continuous counterpart, the constant from the Nečas inequality.For the latter the explicit and asymptotically accurate estimates are proved. The analytic results are illustrated by several numerical experiments.