In this paper, a general technique is developed to enlarge the velocity space ${\rm V}_h^1$ of the unstable -element by adding spaces ${\rm V}_h^2$ such thatfor the extended pair the Babuska-Brezzi condition is satisfied. Examplesof stable elements which can be derived in such a way imply the stability of the well-knownQ2/Q1 -element and the 4Q1/Q1 -element. However, our new elementsare much more cheaper. In particular, we shall see that more than half of theadditional degrees of freedom when switching from the Q 1 to the Q 2 and4Q1 , respectively, element are not necessary to stabilize theQ1/Q1 -element. Moreover, by using the technique of reduced discretizationsand eliminating the additional degrees of freedom we showthe relationship between enlarging the velocity space and stabilized methods. This relationship has been established for triangular elements but was not known for quadrilateral elements. As a result we derive new stabilizedmethods for the Stokes and Navier-Stokes equations. Finally, we showhow the Brezzi-Pitkäranta stabilization and the SUPG method for theincompressible Navier-Stokes equations can be recovered as special cases of the general approach. In contrast to earlier papers we do not restrict ourselves to linearized versions of the Navier-Stokes equations but deal with the full nonlinear case.

The aim of this article is to propose a new method for the grey-level image classification problem. We first present the classicalvariational approach without and with a regularization term in order tosmooth the contours of the classified image. Then we present the generaltopological asymptotic analysis, and we finally introduce its application tothe grey-level image classification problem.

We construct a Galerkin finite element method for the numerical approximation of weak solutions to a coupled microscopic-macroscopic bead-spring model that arises from the kinetic theory of dilute solutionsof polymeric liquids with noninteracting polymer chains. The model consists of the unsteady incompressible Navier–Stokes equations in a bounded domain Ω ⊂ $\mathbb{R}^d$ , d = 2 or 3, for the velocity andthe pressure of the fluid, with an elastic extra-stress tensor as right-hand side in the momentum equation. The extra-stress tensor stems from the random movement of the polymer chains and is defined through the associated probability density function that satisfies a Fokker–Planck type parabolic equation, crucial features of which are the presence of a centre-of-mass diffusion term and a cut-off function $\beta^L(\cdot) :=\min(\cdot,L)$ in the drag and convective terms, where L ≫ 1. We focus on finitely-extensible nonlinear elastic, FENE-type, dumbbell models. We perform a rigorous passage to the limit as the spatial and temporal discretization parameters tend to zero, and show that a (sub)sequence of these finite element approximations converges to a weak solution of this coupled Navier–Stokes–Fokker–Planck system. The passage to the limit is performed under minimal regularity assumptions on the data. Our arguments therefore also provide a new proof of global existence of weak solutions to Fokker–Planck–Navier–Stokes systems with centre-of-mass diffusion and microscopic cut-off. The convergence proof rests on several auxiliary technical results including the stability, in the Maxwellian-weighted H 1 norm, of the orthogonal projector in the Maxwellian-weighted L 2 inner product onto finite element spaces consisting of continuous piecewise linear functions.We establish optimal-order quasi-interpolation error bounds in the Maxwellian-weighted L 2 and H 1 norms,and prove a new elliptic regularity result in the Maxwellian-weighted H 2 norm.

We consider a nonlinear second order elliptic boundary value problem (BVP)in a bounded domain $\Omega\subset {\mathbb R}^N$ witha nonlocal boundary condition. A Dirichlet BC containing an unknown additive constant,accompanied with a nonlocal (integral) Neumann side condition is prescribed at some boundary part Γn . The rest of the boundary is equipped with Dirichlet or nonlinear Robin type BC. The solution is found via linearization. We design a robust and efficient approximation scheme. Error estimates for the linearization algorithm are derived inL 2(Ω),H 1(Ω) andL ∞(Ω) spaces.

This paper is concerned with the dual formulation of the interface problemconsisting of a linear partial differential equation with variable coefficientsin some bounded Lipschitz domain Ω in $\mathbb{R}^n$ (n ≥ 2) and the Laplace equation with some radiation condition in theunbounded exterior domain Ωc := $\mathbb{R}^n\backslash\bar\Omega$ . The two problems are coupled by transmission andSignorini contact conditions on the interface Γ = ∂Ω. The exterior part of theinterface problem is rewritten using a Neumann to Dirichlet mapping (NtD) given in terms of boundary integral operators. The resulting variational formulation becomes a variational inequalitywith a linear operator. Then we treat the corresponding numerical scheme and discuss anapproximation of the NtD mapping with an appropriatediscretization of the inverse Poincaré-Steklov operator. In particular, assuming some abstract approximationproperties and a discrete inf-sup condition, we show unique solvability of the discrete scheme andobtain the corresponding a-priori error estimate. Next, we prove that these assumptions aresatisfied with Raviart-Thomas elements and piecewise constants in Ω, and continuous piecewise linear functions on Γ. We suggest a solver based on a modified Uzawa algorithm and show convergence. Finally we present some numerical results illustrating our theory.

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.

This paper studies a family of finite volume schemes for the hyperbolic scalar conservation law $u_t +\nabla_g \cdotf(x,u)=0$ on a closed Riemannian manifold M.For an initial value in BV(M) we will show that these schemes converge with a $h^{\frac{1}{4}} $ convergence rate towards the entropy solution. When M is 1-dimensional the schemes are TVD and we will show that this improves the convergence rate to $h^{\frac{1}{2}}.$

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 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+.

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 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.

Taking the cue from stabilized Galerkin methods for scalar advection problems, we adaptthe technique to boundary value problems modeling the advection of magnetic fields. Weprovide rigorous a priori error estimates for both fully discontinuouspiecewise polynomial trial functions and -conforming finite elements.

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.

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

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.

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 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.

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.