This paper deals with the problem of numerical approximation in the Cauchy-Dirichlet problem for a scalar conservation law with a flux function having finitely many discontinuities. The well-posedness of this problem was proved by Carrillo [J. Evol. Eq. 3 (2003) 687–705]. Classical numerical methods do not allow us to compute a numerical solution (due to the lack of regularity of the flux). Therefore, we propose an implicit Finite Volume method based on an equivalent formulation of the initial problem. We show the well-posedness of the scheme and the convergence of the numerical solution to the entropy solution of the continuous problem. Numerical simulations are presented in the framework of Riemann problems related to discontinuous transport equation, discontinuous Burgers equation, discontinuous LWR equation and discontinuous non-autonomous Buckley-Leverett equation (lubrication theory).

The numerical solution of ill-posed problems requires suitableregularization techniques. One possible option is to consider timeintegration methods to solve the Showalter differential equationnumerically. The stopping time of the numerical integrator correspondsto the regularization parameter. A number of well-knownregularization methods such as the Landweber iteration or theLevenberg-Marquardt method can be interpreted as variants of theEuler method for solving the Showalter differential equation. Motivated by an analysis of the regularization properties of theexact solution of this equation presented by [U. Tautenhahn, Inverse Problems10 (1994) 1405–1418], we consider a variant of the exponential Euler methodfor solving the Showalter ordinary differential equation. We discuss asuitable discrepancy principle for selecting the step sizes withinthe numerical method and we review the convergence properties of [U. Tautenhahn, Inverse Problems10 (1994) 1405–1418], and of our discrete version [M. Hochbruck et al., Technical Report (2008)].Finally, we present numerical experiments which show that thismethod can be efficiently implemented by using Krylov subspacemethods to approximate the product of a matrix function with avector.

We present numerical evidence for the blow-up of solution for theEuler equations. Our approximate solutions are Taylor polynomials in the timevariable of an exact solution, and we believe that in terms of the exact solution,the blow-up will be rigorously proved.

As an example of a simple constrained geometric non-linear wave equation, we study a numerical approximation of the Maxwell Klein Gordon equation. We consider an existing constraint preserving semi-discrete scheme based on finite elements and prove its convergence in space dimension 2 for initial data of finite energy.

This work represents a first step towards the simulation of the motion of water in a complex hydrodynamic configuration, such asa channel network or a river delta, by means of a suitable“combination” of different mathematicalmodels. In this framework a wide spectrum of space and time scales is involveddue to the presence of physical phenomena ofdifferent nature. Ideally, moving from a hierarchy of hydrodynamic models, one should solvethroughout the whole domain the most complex model (with solution $u_{\rm{fine}}$ ) to accuratelydescribe all the physical features of the problem at hand. In our approach instead,for a user-defined output functional ${\cal F}$ , we aim to approximate,within a prescribed tolerance τ, the value ${\cal F}(u_{\rm{fine}})$ by means of the quantity ${\cal F}(u_{\rm{adapted}})$ , $u_{\rm{adapted}}$ being the so-called adapted solution solving thesimpler models on most of the computational domain while confining the complex ones only on a restricted region. Moving fromthe simplified setting where only two hydrodynamic models, fine and coarse, are considered, weprovide an efficient tool able to automatically selectthe regions of the domain where the coarsemodel rather than the fine one are to be solved, while guaranteeing $|{\cal F}(u_{\rm{fine}})-{\cal F}(u_{\rm{adapted}})|$ below the tolerance τ.This goal is achieved via a suitable a posteriori modeling error analysisdeveloped in the framework of a goal-oriented theory.We extend the dual-based approach provided in [Braack and Ern,Multiscale Model Sim.1 (2003) 221–238],for steady equations to the case of a generic time-dependent problem.Then this analysis is specialized to the case we areinterested in, i.e. the free-surface flows simulation, by emphasizing the crucial issue of the time discretization for both the primal and the dualproblems. Finally, inthe last part of the paper a widespread numerical validation is carried out.

This paper deals with the numerical approximation of mild solutions of elliptic-parabolic equations, relying on the existence results of Bénilan and Wittbold (1996). We introduce a new and simple algorithm based on Halpern's iteration for nonexpansive operators (Bauschke, 1996; Halpern, 1967; Lions, 1977), which is shown to be convergent in the degenerate case, and compare it with existing schemes (Jäger and Kačur, 1995; Kačur, 1999).

Semi-smooth Newton methods for elliptic equations with gradient constraints are investigated. The one- and multi-dimensional cases are treated separately. Numerical examples illustrate the approach and as well as structural features of the solution.

We propose a numerical scheme to compute the motion of a two-dimensional rigid body in a viscous fluid. Our method combines the method of characteristics with a finite element approximation to solve an ALE formulation of the problem. We derive error estimates implying the convergence of the scheme.

Along with the classical requirements on B-splines bases(minimal support, positivity, normalization)we show that it is natural to introduce an additional“end point property". When dealing with multiple knots,this additional property is exactly the appropriate requirementto obtain the poles of nondegenerate splinesas intersections of osculating flats at consecutive knots.

In this paper, we present an abstract framework which describes algebraically the derivation of order conditions independently of the nature of differential equations considered or the type of integrators used to solve them. Our structure includes a Hopf algebra of functions, whose properties are used to answer several questions of prime interest in numerical analysis. In particular, we show that, under some mild assumptions, there exist integrators of arbitrarily high orders for arbitrary (modified) vector fields.

We propose, analyze, and compare several numerical methods for the computation of the deformation of a pressurized martensitic thin film. Numerical results have been obtained for the hysteresis of the deformation as the film transforms reversibly from austenite to martensite.

In vivo visualization of cardiovascular structures ispossible using medical images. However, one has to realize that the resulting 3Dgeometries correspond to in vivo conditions. This entails an internalstress state to be present in the in vivo measured geometry ofe.g. a blood vessel due to the presence of the blood pressure. In orderto correct for this in vivo stress, this paper presents an inverse methodto restore the original zero-pressure geometry of a structure, and to recover thein vivo stress field of the final, loaded structure. The proposedbackward displacement method is able to solve the inverse problem iteratively using fixedpoint iterations, but can be significantly accelerated by a quasi-Newton technique inwhich a least-squares model is used to approximate the inverse of the Jacobian. The hereproposed backward displacement method allows for a straightforward implementation of thealgorithm in combination with existing structural solvers, even if the structural solveris a black box, as only an update of the coordinates of the mesh needs to beperformed.

In a foregoing paper [Sonar, ESAIM: M2AN39 (2005) 883–908] we analyzed the Interpolating Moving Least Squares (IMLS) method due to Lancaster and Šalkauskas with respect to its approximation powers and derived finite difference expressions for the derivatives. In this sequel we follow a completely different approach to the IMLS method given by Kunle [Dissertation (2001)]. As a typical problem with IMLSmethod we address the question of getting admissible results at the boundary by introducing “ghost points”.Most interesting in IMLS methods are the finite difference operators whichcan be computed from different choices of basis functions and weightfunctions. We present a way of deriving these discrete operators inthe spatially one-dimensional case. Multidimensional operatorscan be constructed by simply extending our approach to higher dimensions.Numerical results ranging from 1-d interpolation to the solution of PDEs are given.

Domain decomposition techniques provide a flexible tool for the numerical approximation of partial differential equations. Here, we consider mortar techniques for quadratic finite elements in 3D with different Lagrange multiplier spaces. In particular, we focus on Lagrange multiplier spaces which yield optimal discretization schemes and a locally supported basis for the associated constrained mortar spaces in case of hexahedral triangulations. As a result, standard efficient iterative solvers as multigrid methods can be easily adapted to the nonconforming situation. We present the discretization errors in different norms for linear and quadratic mortar finite elements with different Lagrange multiplier spaces. Numerical results illustrate the performance of our approach.

Space-time approximations of the FitzHugh–Nagumo system of coupled semi-linear parabolicPDEs are examined. The schemes under consideration are discontinuous in time butconforming in space and of arbitrary order. Stability estimates are presented in thenatural energy norms and at arbitrary times, under minimal regularity assumptions.Space-time error estimates of arbitrary order are derived, provided that the naturalparabolic regularity is present. Various physical parameters appearing in the model aretracked and numerical examples are presented.

To filter perturbed local measurements on a random medium, a dynamic model jointly with an observation transfer equation are needed. Some media given by PDE could have a local probabilistic representation by a Lagrangian stochastic process with mean-field interactions. In this case, we define the acquisition process of locally homogeneous medium along a random path by a Lagrangian Markov process conditioned to be in a domain following the path and conditioned to the observations. The nonlinear filtering for the mobile signal is therefore those of an acquisition process contaminated by random errors. This will provide a Feynman-Kac distribution flow for the conditional laws and an N particle approximation with a $\mathcal{O}$ $(\frac{1}{\sqrt{N}})$ asymptotic convergence. An application to nonlinear filtering for 3D atmospheric turbulent fluids will be described.

We are interested in the theoretical study of a spectral problem arising in a physical situation, namely interactions of fluid-solid type structure. More precisely, we study the existence of solutions for a quadratic eigenvalue problem, which describes the vibrations of a system made up of two elastic bodies, where a slip is allowed on their interface and which surround a cavity full of an inviscid and slightly compressible fluid. The problem shall be treated like a generalized eigenvalue problem. Thus by using some functional analysis results, we deduce the existence of solutions and prove a spectral asymptotic behavior property, which allows us to compare the spectrum of this coupled model and the spectrum associated to the problem without transmission between the fluid-solid media.

The equilibrium configurations of a one-dimensional variational model thatcombines terms expressing the bulk energy of a deformable crystal and itssurface energy are studied. After elimination of the displacement, theproblem reduces to the minimization of a nonconvex and nonlocal functional ofa single function, the thickness. Depending on a parameter which strengthensone of the terms comprising the energy at the expense of the other, it isshown that this functional may have a stable absolute minimum or only aminimizing sequence in which the term corresponding to the bulk energy isforced to zero by the production of a crack in the material.

This paper deals with the numerical study of a nonlinear, strongly anisotropic heatequation. The use of standard schemes in this situation leads to poor results, due to thehigh anisotropy. An Asymptotic-Preserving method is introduced in this paper, which issecond-order accurate in both, temporal and spacial variables. The discretization in timeis done using an L-stable Runge−Kutta scheme. The convergence of the method is shown to beindependent of the anisotropy parameter , andthis for fixed coarse Cartesian grids and for variable anisotropy directions. The contextof this work are magnetically confined fusion plasmas.

A simplified stochastic Hookean dumbbells model arising from viscoelastic flows is considered, the convective terms being disregarded. A finite element discretization in space is proposed. Existence of the numerical solution is proved for small data, so as a priori error estimates,using an implicit function theorem and regularity results obtained in [Bonito et al., J. Evol. Equ.6 (2006) 381–398] for the solution of the continuous problem. A posteriori error estimates are also derived.Numerical results with small time steps and a large number of realizations confirm theconvergence rate with respect to the mesh size.