Since matrix compression has paved the way for discretizing the boundary integralequation formulations of electromagnetics scattering on very fine meshes, preconditionersfor the resulting linear systems have become key to efficient simulations. Operatorpreconditioning based on Calderón identities has proved to be a powerful device fordevising preconditioners. However, this is not possible for the usual first-kind boundaryformulations for electromagnetic scattering at general penetrable composite obstacles. Wepropose a new first-kind boundary integral equation formulation following the reasoningemployed in [X. Clayes and R. Hiptmair, Report 2011-45, SAM, ETH Zürich (2011)] foracoustic scattering. We call it multi-trace formulation, because itsunknowns are two pairs of traces on interfaces in the interior of the scatterer. We give acomprehensive analysis culminating in a proof of coercivity, and uniqueness and existenceof solution. We establish a Calderón identity for the multi-trace formulation, which formsthe foundation for operator preconditioning in the case of conforming Galerkin boundaryelement discretization.

Motivated by the development of efficient Monte Carlo methodsfor PDE models in molecular dynamics,we establish a new probabilistic interpretation of a family of divergence formoperators with discontinuous coefficients at the interfaceof two open subsets of $\mathbb{R}^d$ . This family of operators includes the case of thelinearized Poisson-Boltzmann equation used tocompute the electrostatic free energy of a molecule.More precisely, we explicitly construct a Markov process whoseinfinitesimal generator belongs to this family, as the solution of a SDEincluding a non standard local time term related to the interfaceof discontinuity. We then prove an extendedFeynman-Kac formula for the Poisson-Boltzmann equation.This formula allows us to justifyvarious probabilistic numerical methods toapproximate the free energy of a molecule.We analyse the convergence rate of these simulation procedures andnumerically compare them on idealized molecules models.

Low order edge elements are widely used for electromagnetic field problems. Higher order edge approximations are receiving increasing interest but their definition become rather complex.In this paper we propose a simple definition for Whitney edge elements of polynomial degree higher than one. We give a geometrical localization of all degrees of freedom over particular edges and provide a basis for these elements on simplicial meshes.As for Whitney edge elements of degree one, the basis is expressedonly in terms of the barycentric coordinates of the simplex.

This paper is concerned with the computation of 3D vertex singularities of anisotropic elastic fields with Dirichlet boundary conditions, focusing on the derivation of error estimates for a finite element method on graded meshes. The singularities are described by eigenpairs of a corresponding operator pencil on spherical polygonal domains. The main idea is to introduce a modified quadratic variational boundary eigenvalue problem which consists of two self-adjoint, positive definite sesquilinear forms and a skew-Hermitean form. This eigenvalue problem is discretized by a finite element method on graded meshes. Based on regularity results for the eigensolutions estimates for the finite element error are derived both for the eigenvalues and the eigensolutions. Finally, some numerical results are presented.

We consider the identification of a distributed parameter in an ellipticvariational inequality. On the basis of an optimal control problemformulation, the application of a primal-dual penalizationtechnique enables us to prove the existenceof multipliers giving a first order characterization of the optimal solution.Concerning the parameter we consider differentregularity requirements. For the numerical realization we utilize a complementarity function,which allows us to rewrite the optimality conditions as a set of equalities.Finally, numerical results obtained from a least squares type algorithmemphasize the feasibility of our approach.

In this paper we study different algorithms for backward stochastic differential equations (BSDE in short) basing on random walk framework for 1-dimensional Brownian motion. Implicit and explicit schemes for both BSDE and reflected BSDE are introduced. Then we prove the convergence of different algorithms and present simulation results for different types of BSDEs.

An optimal shape control problem for the stationary Navier-Stokessystem is considered. An incompressible, viscous flow in atwo-dimensional channel is studied to determine the shape of part ofthe boundary that minimizes the viscous drag. Theadjoint method and the Lagrangian multiplier method are used to derivethe optimality system for the shapegradient of the design functional.

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.

In this paper we introduce and analyze some non-overlapping multiplicative Schwarz methods for discontinuous Galerkin (DG) approximations of elliptic problems. The construction of the Schwarz preconditioners is presented in a unified framework for a wide class of DG methods.For symmetric DG approximations we provide optimal convergence bounds for the corresponding error propagation operator, and we show that the resulting methods can be accelerated by using suitable Krylov space solvers. A discussion on the issue of preconditioning non-symmetric DG approximations of elliptic problems is also included. Extensive numerical experiments to confirm the theoretical results and to assess the robustness and the efficiency of the proposed preconditioners are provided.

In this work we study a fully discrete mixed scheme, based on continuous finite elementsin space and a linear semi-implicit first-order integration in time, approximating anEricksen–Leslie nematic liquid crystal model by means of aGinzburg–Landau penalized problem. Conditional stability of this schemeis proved via a discrete version of the energy law satisfied by thecontinuous problem, and conditional convergence towards generalized Young measure-valuedsolutions to the Ericksen–Leslie problem is showed when the discreteparameters (in time and space) and the penalty parameter go to zero at the same time.Finally, we will show some numerical experiences for a phenomenon of annihilation ofsingularities.

We present a domain decomposition theory on an interface problemfor the linear transport equation between a diffusive and a non-diffusive region.To leading order, i.e. up to an error of the order of the mean free path in thediffusive region, the solution in the non-diffusive region is independent of thedensity in the diffusive region. However, the diffusive and the non-diffusive regionsare coupled at the interface at the next order of approximation. In particular, ouralgorithm avoids iterating the diffusion and transport solutions as is done in mostother methods — see for example Bal–Maday (2002). Our analysis is based instead on an accurate description of the boundarylayer at the interface matching the phase-space density of particles leaving thenon-diffusive region to the bulk density that solves the diffusion equation.

This work is concerned with the flow of a viscousplastic fluid. We choose a model of Bingham typetaking into account inhomogeneous yield limit of thefluid, which is well-adapted in the description oflandslides. After setting the generalthreedimensional problem, the blocking property isintroduced. We then focus on necessary andsufficient conditions such that blocking of the fluidoccurs.The anti-plane flow intwodimensional andonedimensional cases is considered.A variational formulation in terms of stresses isdeduced. More fine properties dealing with localstagnant regions as well as local regions where thefluid behaves like a rigid body are obtained indimension one.

We introduce a family of new second-order Godunov-type central schemes for one-dimensional systems of conservation laws. They are a less dissipative generalization of the central-upwind schemes,proposed in [A. Kurganov et al., submitted to SIAM J. Sci. Comput.], whose construction is based on the maximal one-sided local speeds of propagation. We also present a recipe, which helps to improve the resolution of contact waves. This is achieved by using the partial characteristic decomposition, suggested by Nessyahu and Tadmor [J. Comput. Phys.87 (1990) 408-463], which is efficiently applied in the context of the new schemes.The method is tested on the one-dimensional Euler equations, subject to differentinitial data, and the results are compared to the numericalsolutions, computed by other second-order central schemes.The numerical experiments clearly illustrate the advantages of theproposed technique.

The incompressible Navier-Stokes problem is discretized in time by the two-step backward differentiation formula. Error estimates are proved under feasible assumptions on the regularity of the exact solution avoiding hardly fulfillable compatibility conditions. Whereas the time-weighted velocity error is of optimal second order, the time-weighted error in the pressure is of first order. Suboptimal estimates are shown for a linearisation. The results cover both the two- and three-dimensional case.

In this paper, we consider linear ordinary differential equations originating in electronic engineering, which exhibit exceedingly rapid oscillation. Moreover, the oscillation model is completely different from the familiar framework of asymptotic analysis of highly oscillatory integrals. Using a Bessel-function identity, we expand the oscillator into asymptotic series, and this allows us to extend Filon-type approach to this setting. The outcome is a time-stepping method that guarantees high accuracy regardless of the rate of oscillation.

The Method of Fundamental Solutions (MFS) is a boundary-typemeshless method for the solution of certain elliptic boundaryvalue problems. In this work, we investigate the properties of thematrices that arise when the MFS is applied to theDirichlet problem for Laplace's equation in a disk. In particular,we study the behaviour of the eigenvalues of these matrices andthe cases in which they vanish. Based on this, we propose amodified efficient numerical algorithm for the solution of theproblem which is applicable even in the cases when the MFS matrixmight be singular. We prove the convergence of the method foranalytic boundary data and perform a stability analysis of the methodwith respect to the distance of the singularities from the originand the number of degrees of freedom. Finally, wetest the algorithm numerically.

In the present work we investigate the numerical simulation of liquid-vapor phase changein compressible flows. Each phase is modeled as a compressible fluid equipped with its ownequation of state (EOS). We suppose that inter-phase equilibrium processes in the mediumoperate at a short time-scale compared to the other physical phenomena such as convectionor thermal diffusion. This assumption provides an implicit definition of an equilibriumEOS for the two-phase medium. Within this framework, mass transfer is the result of localand instantaneous equilibria between both phases. The overall model is strictlyhyperbolic. We examine properties of the equilibrium EOS and we propose a discretizationstrategy based on a finite-volume relaxation method. This method allows to cope with theimplicit definition of the equilibrium EOS, even when the model involves complex EOS’s forthe pure phases. We present two-dimensional numerical simulations that shows that themodel is able to reproduce mechanism such as phase disappearance and nucleation.

Galerkin discretizations of integral equations in $\mathbb{R}^{d}$ requirethe evaluation of integrals $I = \int_{S^{(1)}}\int_{S^{(2)}}g(x,y){\rm d}y{\rm d}x$ where S (1),S (2) are d-simplices and g has a singularityat x = y. We assume that g is Gevrey smooth for x $\ne$ y andsatisfies bounds for the derivatives which allow algebraic singularitiesat x = y. This holds for kernel functions commonly occurring in integralequations. We construct a family of quadrature rules $\mathcal{Q}_{N}$ usingN function evaluations of g which achieves exponential convergence|I – $\mathcal{Q}_{N}$ | ≤C exp(–r N γ ) with constants r, γ > 0.

We study here the water waves problem for uneven bottoms in a highly nonlinear regime wherethe small amplitude assumption of the Korteweg-de Vries (KdV) equation is enforced. It is knownthat, for such regimes, a generalization of the KdV equation (somehow linked to the Camassa-Holm equation) can be derived and justified [Constantin and Lannes, Arch. Ration. Mech. Anal. 192 (2009) 165–186] when the bottom isflat. We generalize here this resultwith a new class of equations taking into account variable bottom topographies. Of course, many variable depth KdV equations existing in the literature are recovered as particular cases.Various regimes for the topography regimes are investigated and we prove consistency of these models, as well as a full justificationfor some of them. We also study the problem of wave breaking for our newvariable depth and highly nonlinear generalizations of the KdV equations.

In this paper we analyze the effect of introducing a numerical integration in the piecewise linear finite element approximation of the Steklov eigenvalue problem. We obtain optimal order error estimates for the eigenfunctions when this numerical integration is used and we prove that, for singular eigenfunctions, the eigenvalues obtained using this reduced integration are better approximations than those obtained using exact integration when the mesh size is small enough.