For scalar conservation laws in one space dimension with a flux function discontinuous inspace, there exist infinitely many classes of solutions which are L1 contractive.Each class is characterized by a connection (A,B) which determines the interface entropy. Forsolutions corresponding to a connection (A,B), there exists convergent numerical schemesbased on Godunov or Engquist−Osher schemes. The natural question is how to obtain schemes,corresponding to computationally less expensive monotone schemes like Lax−Friedrichs etc., usedwidely in applications. In this paper we completely answer this question for more general(A,B)stable monotone schemes using a novel construction of interface flux function. Then fromthe singular mapping technique of Temple and chain estimate of Adimurthi and Gowda, weprove the convergence of the schemes.

We discuss a parallel implementation of the domaindecomposition method based on the macro-hybrid formulationof a second order ellipticequation and on an approximation by the mortar element method. The discretization leads to an algebraic saddle- point problem. An iterative method with a block- diagonalpreconditioner is used for solving the saddle- point problem. A parallel implementation of the method is emphasized. Finally the results of numerical experiments are presented.

We consider a two-dimensional quantum waveguide composed of two semi-strips of width 1and 1 − ε, where ε > 0 is a small real parameter,i.e. the waveguide is gently converging. The width of the junction zonefor the semi-strips is 1 + O(√ε). We will present a sufficient condition for the existence of a weaklycoupled bound state below π2, the lower bound of thecontinuous spectrum. This eigenvalue in the discrete spectrum is unique and itsasymptotics is constructed and justified whenε → 0+.

We introduce a piecewise P2-nonconforming quadrilateralfinite element. First, we decompose a convex quadrilateral into the union of fourtriangles divided by its diagonals. Then the finite element space is defined by the set ofall piecewise P2-polynomials that are quadratic in eachtriangle and continuously differentiable on the quadrilateral. The degrees of freedom(DOFs) are defined by the eight values at the two Gauss points on each of the four edgesplus the value at the intersection of the diagonals. Due to the existence of one linearrelation among the above DOFs, it turns out the DOFs are eight. Global basis functions aredefined in three types: vertex-wise, edge-wise, and element-wise types. The correspondingdimensions are counted for both Dirichlet and Neumann types of elliptic problems. Forsecond-order elliptic problems and the Stokes problem, the local and global interpolationoperators are defined. Also error estimates of optimal order are given in both brokenenergy and L2(Ω) norms. The proposed elementis also suitable to solve Stokes equations. The element is applied to approximate eachcomponent of velocity fields while the discontinuousP1-nonconforming quadrilateral element is adopted toapproximate the pressure. An optimal error estimate in energy norm is derived. Numericalresults are shown to confirm the optimality of the presented piecewiseP2-nonconforming element on quadrilaterals.

We examine an elliptic optimal control problem with control and state constraints inℝ3. An improved error estimate of 𝒪(hs) with 3/4 ≤ s ≤ 1 − ε is proven for a discretisationinvolving piecewise constant functions for the control and piecewise linear for the state.The derived order of convergence is illustrated by a numerical example.

Cell-centered and vertex-centered finite volume schemes for the Laplace equationwith homogeneous Dirichlet boundary conditionsare considered on a triangular mesh and on the Voronoi diagram associated to its vertices.A broken P 1 function is constructed from the solutions of both schemes.When the domain is two-dimensional polygonal convex,it is shown that this reconstructionconverges with second-order accuracy towards the exact solution in the L 2 norm,under the sufficient condition that the right-hand side of the Laplace equation belongs to H 1(Ω).

Singularly perturbed reaction-diffusionproblems exhibit in general solutions with anisotropic features,e.g. strong boundary and/or interior layers.This anisotropy is reflected in a discretization by using mesheswith anisotropic elements. The quality of the numerical solutionrests on the robustness of the a posteriori error estimator withrespect to both, the perturbation parameters of the problemand the anisotropy of the mesh. The equilibrated residual method has been shown to provide one of the most reliable error estimates for the reaction-diffusion problem. Its modification suggested byAinsworth and Babuška has been proved to be robust for the case of singular perturbation. In the present work we investigate the modified method on anisotropic meshes. The method in the form of Ainsworth and Babuška is shown here to fail on anisotropic meshes. We suggest a new modification based on the stretching ratios of the mesh elements. The resulting error estimator is equivalent tothe equilibrated residual method in the case of isotropic meshesand is proved to be robust on anisotropic meshes as well. Among others, the equilibrated residual method involves the solution of an infinite dimensional local problem on each element. In practical computations an approximate solution to this local problem was successfully computed. Nevertheless, up to now no rigorous analysis has been done showing the appropriateness of any computable approximation. This demands special attention since an improper approximate solution to the local problem can be fatal for the robustness of the whole method. In the present work we provide one of the desired approximations. We prove that the method is not affected by the approximate solution of the local problem.

We prove the optimal convergence of a discontinuous-Galerkin-based immersed boundary method introduced earlier [Lew and Buscaglia, Int. J. Numer. Methods Eng.76 (2008) 427–454]. By switching to a discontinuous Galerkin discretization near the boundary, this method overcomes the suboptimal convergence rate that may arise in immersed boundary methods when strongly imposing essential boundary conditions. We consider a model Poisson's problem with homogeneous boundary conditions over two-dimensional C 2-domains. For solution in Hq for q > 2, we prove that the method constructed with polynomials of degree one on each element approximates the function and its gradient with optimal orders h 2 and h, respectively. When q = 2, we have h 2-ε and h 1-ε for any ϵ > 0 instead. To this end,we construct a new interpolant that takes advantage of the discontinuities in the space, since standard interpolation estimates lead here to suboptimal approximation rates. The interpolation error estimate is based on proving an analog to Deny-Lions' lemma for discontinuous interpolants on a patch formed by the reference elements of any element and its three face-sharing neighbors. Consistency errors arising due to differences between the exact and the approximate domains are treated using Hardy's inequality together with more standard results on Sobolev functions.

In this short note we correct a conceptual error in theheuristic derivation of a kinetic equation used for thedescription of a one-dimensional granular medium in the socalled quasi-elastic limit, presented by the same authors inreference[1]. The equation we derived is however correct so that,the rigorous analysis on this equation, which constituted themain purpose of that paper, remains unchanged.

We prove existence (uniqueness is easy) of a weak solution to a boundary value problem for an equation like $(v-1)^+_t = v_{xx} + F(v)_x$ where the function $F: \Bbb R\rightarrow\Bbb R$ is onlysupposed to be locally lipschitz continuous. In order to replace the lack of compactness in t on v<1, we use nonlinear semigroup theory.

We present in this paper the formal passage from a kinetic model to the incompressible Navier−Stokes equations for a mixture of monoatomic gases with different masses. The starting point of this derivation is the collection of coupled Boltzmann equations for the mixture of gases. The diffusion coefficients for the concentrations of the species, as well as the ones appearing in the equations for velocity and temperature, are explicitly computed under the Maxwell molecule assumption in terms of the cross sections appearing at the kinetic level.

We consider linear elliptic problems with variable coefficients, which may sharply change values and have a complex behavior in the domain. For these problems, a new combined discretization-modeling strategy is suggested and studied. It uses a sequence of simplified models, approximating the original one with increasing accuracy. Boundary value problems generated by these simplified models are solved numerically, and the approximation and modeling errors are estimated by a posteriori estimates of functional type. An efficient numerical strategy is based upon balancing the modeling and discretization errors, which provides an economical way of finding an approximate solution with an a priori given accuracy. Numerical tests demonstrate the reliability and efficiency of this combined modeling-discretization method.

In this article we implement different numerical schemes to simulate the Schrödinger-Debye equations that occur in nonlinear optics. Since the existence of blow-up solutions is an open problem, we try to compute such solutions. The convergence of the methods is proved and simulations seem indeed to show that for at least small delays self-focusing solutions may exist.

In this paper we consider the Maxwell resolvent operator and its finite elementapproximation. In this framework it is natural the use of the edge elementspaces and to impose the divergence constraint in a weaksense with the introduction of a Lagrange multiplier, followingan idea by Kikuchi [14].We shall review some of the known properties for edge elementapproximations and prove some new result. In particular we shall prove auniform convergence in the L 2 norm for the sequence of discrete operators.These results, together with a general theory introduced by Brezzi, Rappaz andRaviart [8], allow an immediate proof of convergence for thefinite element approximation of the time-harmonicMaxwell system.

Based on estimates for the KdV equation in analyticGevrey classes, a spectral collocation approximation ofthe KdV equation is proved to converge exponentially fast.

A numerically inexpensive globalization strategy of sequential quadratic programming methods (SQP-methods) for control of the instationary Navier Stokes equations is investigated. Based on the proper functional analytic setting a convergence analysis for the globalized methodis given. It is argued that the a priori formidable SQP-step can be decomposed into linear primal and linear adjoint systems, which is amenable for existing CFL-software. A report on a numerical test demonstrates the feasibility of the approach.

Recently, adaptive wavelet strategies for symmetric, positive definite operators have been introduced that were proven to converge.This paper is devoted to the generalization to saddle point problems which are also symmetric, but indefinite. Firstly, we investigate a posteriori error estimates and generalize the known adaptive wavelet strategy to saddle point problems. The convergence of this strategy for elliptic operators essentially relies on the positive definite character of the operator. As an alternative, we introduce an adaptive variant of Uzawa's algorithm and prove its convergence. Secondly, we derive explicit criteria for adaptively refined wavelet spaces in order to fulfill the Ladyshenskaja-Babuška-Brezzi (LBB) condition and to be fully equilibrated.

In this work we consider the dual-primal Discontinuous Petrov–Galerkin (DPG) method for the advection-diffusion model problem. Since in the DPG method both mixed internal variables are discontinuous,a static condensation procedure can becarried out, leading to a single-field nonconformingdiscretization scheme. For this latter formulation,we propose a flux-upwind stabilization technique to deal with the advection-dominated case. The resulting scheme is conservative and satisfies a discrete maximum principle under standard geometrical assumptions on the computational grid. A convergence analysis isdeveloped, proving first-order accuracy of the method in a discrete H 1-norm, and the numerical performance of the scheme is validated on benchmark problems with sharp internal and boundary layers.

This article presents the derivation of a semi-classical model of electromagnetic-wave propagation in a non centro-symmetric crystal. It consists of Maxwell's equations for the wave field coupled with a version of Bloch's equations which takes fully into account the discrete symmetry group of the crystal. The model is specialized in the case of a KDP crystal for which information about the dipolar moments at the Bloch level can be recovered from the macroscopic dispersion properties of the material.

We discuss new MUSCL reconstructions to approximate the solutions of hyperbolic systems of conservations laws on 2D unstructured meshes. To address such an issue, we write two MUSCL schemes on two overlapping meshes. A gradient reconstruction procedure is next defined by involving both approximations coming from each MUSCL scheme. This process increases the number of numerical unknowns, but it allows to reconstruct very accurate gradients. Moreover a particular attention is paid on the limitation procedure to enforce the required robustness property. Indeed, the invariant region is usually preserved at the expense of a more restrictive CFL condition. Here, we try to optimize this condition in order to reduce the computational cost.