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.

In this paper, a Dirichlet-Neumann substructuring domaindecomposition method is presented for a finite elementapproximation to the nonlinear Navier-Stokes equations. It isshown that the Dirichlet-Neumann domain decomposition sequenceconverges geometrically to the true solution provided the Reynoldsnumber is sufficiently small. In this method, subdomain problemsare linear. Other version where the subdomain problems are linearStokes problems is also presented.

In order to describe a solid which deforms smoothly in some region, but non smoothly in some other region, many multiscale methods have recently been proposed. They aim at coupling an atomistic model (discrete mechanics) with a macroscopic model (continuum mechanics). We provide here a theoretical ground for such a coupling in a one-dimensional setting. We briefly study the general case of a convex energy, and next concentrate on a specific example of a nonconvex energy, the Lennard-Jones case. In the latter situation, we prove that the discretization needs to account in an adequate way for the coexistence of a discrete model and a continuous one. Otherwise, spurious discretization effects may appear. We provide a numerical analysis of the approach.

This paper addresses some results on the development of an approximate methodfor computing the acoustic field scattered by a three-dimensional penetrable object immersed into an incompressiblefluid. The basic idea of the method consists in using on-surface differentialoperators that locally reproduce the interior propagation phenomenon. This approach leads tointegral equation formulations with a reduced computational cost compared to standard integral formulations couplingboth the transmitted and scattered waves. Theoreticalaspects of the problem and numerical experiments are reported to analyze the efficiency ofthe method and precise its validity domain.

We study the propagation of electromagnetic waves in a guide the section of which is a thin annulus. Owing to the presence of a small parameter, explicit approximations of the TM and TE eigenmodes are obtained. The cases of smooth and non smooth boundaries are presented.

The calculation of sound generation and propagation in low Mach number flows requires serious reflections on the characteristics of the underlying equations. Although the compressible Euler/Navier-Stokes equations cover all effects, an approximation via standard compressible solvers does not have the ability to represent acoustic waves correctly. Therefore, different methods have been developed to deal with the problem. In this paper, three of them are considered and compared to each other. They are the Multiple Pressure Variables Approach (MPV), the Expansion about Incompressible Flow (EIF) and a coupling method via heterogeneous domain decomposition. In the latter approach, the non-linear Euler equations are used in a domain as small as possible to cover the sound generation, and the locally linearized Euler equations approximated with a high-order scheme are used in a second domain to deal with the sound propagation. Comparisons will be given in construction principles as well as implementational effort and computational costs on actual numerical examples.

We perform a complete study of the truncation error of the Gegenbauer series. This series yields an expansion of the Green kernel of the Helmholtz equation, $\frac{ {\rm e}^{i |\vec{u}-\vec{v}|}}{4 \pi i |\vec{u}-\vec{v}|}$, which is the core of the Fast Multipole Method for the integral equations. We consider the truncated series where the summation is performed over the indices $\ell \le L$. We prove that if $v = |\vec{v}|$ is large enough, the truncated series gives rise to an error lower than ϵ as soon as L satisfies $L+\frac{1}{2} \simeq v + C W^{\frac{2}{3}}(K(\alpha) \epsilon^{-\delta} v^\gamma )\, v^{\frac{1}{3}}$ where W is the Lambert function, $K(\alpha)$ depends only on $\alpha=\frac{|\vec{u}|}{|\vec{v}|}$ and $C\,, \delta, \, \gamma$ are pure positive constants. Numerical experiments show that this asymptotic is optimal. Those results are useful to provide sharp estimates of the error in the fast multipole method for scattering computation.

The 2D-Signorini contact problem with Tresca and Coulomb friction is discussed in infinite-dimensional Hilbert spaces. First, the problem with given friction (Tresca friction) is considered. It leads to a constraint non-differentiable minimization problem. By means of the Fenchel duality theorem this problem can be transformed into a constrained minimization involving a smooth functional. A regularization technique for the dual problem motivated by augmented Lagrangians allows to apply an infinite-dimensional semi-smooth Newton method for the solution of the problem with given friction. The resulting algorithm is locally superlinearly convergent and can be interpreted as active set strategy. Combining the method with an augmented Lagrangian method leads to convergence of the iterates to the solution of the original problem. Comprehensive numerical tests discuss, among others, the dependence of the algorithm's performance on material and regularization parameters and on the mesh. The remarkable efficiency of the method carries over to the Signorini problem with Coulomb friction by means of fixed point ideas.

The results of a workshop concerning the numericalsimulation of the liquid flow around a hydrofoil in non-cavitating andcavitating conditions are presented. This workshop was part of theconference “Mathematical and Numerical aspects of Low Mach NumberFlows” (2004) and was aimed to investigate the capabilities ofdifferent compressible flow solvers for the low Mach number regime and forflows in which incompressible and supersonic regions aresimultaneously present. Different physical models of cavitatingphenomena are also compared. The numerical results are validatedagainst experimental data.

This paper is devoted to the numerical simulation of wavebreaking. It presents the results of a numerical workshop that washeld during the conference LOMA04. The objective is to compareseveral mathematical models (compressible or incompressible) andassociated numerical methods to compute the flow field during awave breaking over a reef. The methods will also be compared withexperiments.

There are very few reference solutions in the literature onnon-Boussinesq natural convection flows. We propose here a testcase problem which extends the well-known De Vahl Davisdifferentially heated square cavity problem to the case of largetemperature differences for which the Boussinesq approximation isno longer valid. The paper is split in two parts: in this firstpart, we propose as yet unpublished reference solutions for casescharacterized by a non-dimensional temperature difference of 0.6,Ra = 106 (constant property and variable property cases) andRa = 107 (variable property case). These reference solutions wereproduced after a first international workshop organized by CEA andLIMSI in January 2000, in which the above authors volunteered toproduce accurate numerical solutions from which the presentreference solutions could be established.

In the second part of the paper, we compare the solutions producedin the framework of the conference “Mathematical and numericalaspects of low Mach number flows” organized by INRIA and MAB inPorquerolles, June 2004, to the reference solutions described inPart 1. We make some recommendations on how to produce goodquality solutions, and list a number of pitfalls to be avoided.