Mixed Discontinuous Galerkin Finite Element Method for the Biharmonic Equation

In this paper, we first split the biharmonic equation Δ² u=f with nonhomogeneous essential boundary conditions into a system of two second order equations by introducing an auxiliary variable v=Δu and then apply an hp-mixed discontinuous Galerkin method to the resulting system. The unknown approximation v _h of v can easily be eliminated to reduce the discrete problem to a Schur complement system in u _h , which is an approximation of u. A direct approximation v _h of v can be obtained from the approximation u _h of u. Using piecewise polynomials of degree p≥3, a priori error estimates of u−u _h in the broken H ¹ norm as well as in L ² norm which are optimal in h and suboptimal in p are derived. Moreover, a priori error bound for v−v _h in L ² norm which is suboptimal in h and p is also discussed. When p=2, the preset method also converges, but with suboptimal convergence rate. Finally, numerical experiments are presented to illustrate the theoretical results. Supported by DST-DAAD (PPP-05) project.     

H 1-Galerkin Mixed Finite Element Method for the Regularized Long Wave Equation

In this paper, an H¹-Galerkin mixed finite element method is proposed for the 1-D regularized long wave (RLW) equation u_t+u_x+uu_x−δu_xxt=0. The existence of unique solutions of the semi-discrete and fully discrete H¹-Galerkin mixed finite element methods is proved, and optimal error estimates are established. Our method can simultaneously approximate the scalar unknown and the vector flux effectively, without requiring the LBB consistency condition. Finally, some numerical results are provided to illustrate the efficacy of our method.     

Numerical solution of parabolic problems by the generalized Crank-Nicolson scheme

A Galerkin procedure is used to obtain a semi-discretization for parabolic equations such as the heat equationu _t =t _xx . The time variable being left continuous, the higher order approximation thus obtained for the space variable is then matched by a higher order discretization of the system of ordinary differential equations that results. Specifically we choose the Padé (2,2), and show how complex factorization it can be practically used. Moreover we prove that the operation count is 0 (h ⁻²) as compared to 0(h ⁻³) with the classical Crank-Nicolson. Numerical calculations are available. This work was supported by the Office of Naval Research, and the Lebanese Council for Scientific Research.     

Functional expansions for nonlinear discrete-time systems

Given an input-output map associated with a nonlinear discrete-time state equationx(t + 1) =f(x(t);u(t)) and a nonlinear outputy(t) =h(x(t)), we present a method for obtaining a “discrete Volterra series” representation of the outputy(t) in terms of the controlsu(0), ...,u(t − 1). The proof is based on Taylor-type expansions of the iterated composition of analytic functions. It allows us to make an explicit construction of each kernel, that is, each coefficient of the series expansion ofy(t) in powers of the controls. This is achieved by making use of successive directional derivatives associated with a family of vector fields which are deduced from the discrete state equations. We discuss the use of these vector fields for the analysis and control of nonlinear discrete-time systems. This work was carried out while D. Normand-Cyrot was working at the I.A.S.I. (from March to October 1984) and with the financial support of the Italian C.N.R. (Consiglio Nazionale delle Ricerche).     

A penalty method for the approximation of projections over cones in Hilbert spaces

Given a closed convex coneK in a Hilbert spaceH and a vectoru ₀ ∈H, a penalty method is built up in order to approximate the projection ofu ₀ over the polar coneK ^* ofK, without making use of the inverse transform of the canonical mapping ofH into its dual spaceH′. Such method is outlined in n⁰ 1, 2. In n⁰3 a complete analysis of the errors of the method is explained. In n⁰4 the method is applied to find error bounds for the numerical approximation of the projection ofu ₀ onK.     

Sur les cycles élémentaires dans les graphes et hypergraphesk-chromatiques

Résumé Le but de ce travail est celui d'obtenir une borne inférieure de la longueur des plus longues cycles élémentaires pour un graphe ou un hypergraphe de nombre chromatique donné. Ainsi on démontre que tout graphe qui n'a pas (r+1)-cliques, de degré minimal ≥d (ou de nombre chromatiqued+1) contient un cycle élémentaire de longueur ≥rd/(r−1) et tout hypergrapheH de nombre chromatique χ(H)=k contient un cycle élémentaire de longueur ≥k. On obtient aussi que tout grapheG de nombre chromatique γ(G)=k≥3 qui n'a pas de triangles contient un cycle élémentaire de longueur ≥2k−1, résultat qui est généralisé sous la forme suivante: Si le grapheG de degré minimal ≥d est 2-connexe et ne contient pas de triangles, alorsG=K _d,d′ oùd′≥d, ouG contient un cycle élémentaire de longueur ≥2d+1. On déduit que tout grapheG avec γ(G)=k≥3, qui n'a pask-cliques contient un cycle élémentaire de longueur ≥k+2, cette borne étant atteinte.

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The aim of this paper is that to obtain a lower bound of the length of the longest elementary cycles of ak-chromatic graph or hypergraph. It is proved that any graph without (r+1)-cliques, having the minimal degree of its vertices ≥d (or the chromatic number equal tod+1) has an elementary cycle of length ≥rd/(r−1) and any hypergraphH of chromatic number χ(H)=k has an elementary cycle of length ≥k. It is obtained also that any graphG without triangles, having chromatic number γ(G)=k≥3 contains an elementary cycle of length ≥2k−1. This result is generalized in the following way: IfG is 2-connected, has minimal degree ≥d and contains no triangle, thenG=K _d,d′ withd′≥d orG has an elementary cycle of length ≥2d+1. It is derived that any graphG with γ(G)=k≥3 which does not containk-cliques has an elementary cycle of length ≥k+2, this bound being attained.

     

Error estimates for two-phase stefan problems in several space variables,I: Linear boundary conditions

The enthalpy formulation of two-phase Stefan problems, with linear boundary conditions, is approximated by C⁰-piecewise linear finite elements in space and backward-differences in time combined with a regularization procedure. Error estimates of L²-type are obtained. For general regularized problems an order ε^1/2 is proved, while the order is shown to be ε for non-degenerate cases. For discrete problems an order h²ε⁻¹+h+τε^−1/2+τ^2/3 is obtained. These orders impose the relations ε∼τ∼h^4/3 for the general case and ε∼h∼τ^2/3 for non-degenerate problems, in order to obtain rates of convergence h^2/3 or h respectively. Besides, an order h|log h|+τ^1/2 is shown for finite element meshes with certain approximation property. Also continuous dependence of discrete solutions upon the data is proved. This work was supported by the “Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET)” of Argentina.     

Il metodo di Treffiz-Fichera per i semigruppi. potenziale di tipo potenza

A variation of the Trefftz-Fichera method is presented to compute lower bounds for the eigenvalues of a positive self-adjoint operator with discrete spectrum with grow at least in a logarithmic way as the index diverges. As suggested by Barnes et al. [2] to compute ground state, the semigroupe ^−βH, β>0, is used rather than the iterated resolvent(H+β) ⁻ⁿ,n=1,2,... As an example, the method is applied to the operatorH=−Δ+|x|^γ. inL ²(R), 1≤γ≤4.      

Optimal numerical differentiation usingN function evaluations

An optimal choice ofu for approximating thed ^th derivative,d=1,2, of a real valued function of a real variable by a difference quotient of the formh ^−d ∑ _i=1 ⁿ w _i f(x+u _i h) is given. Ifd=1 andn is odd, or ifd=2 andn is even, this choiceu turns out to be surprisingly asymmetric.     

Sample complexity of worst-case H-identification

We show that the sample complexity of qorst-case H^∞-identification is of order n², by proving that the minimal length of a fractional H^∞-cover for C_n, regarded as the linear space of complex-valued sequences of length n, is of order n². A unit vector u in ^∞ is a fractional H^∞-cover for C_n if for some

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for all rh ε C_n, where is the z-transform of h. We also give similar results for real-valued sequences.     

A nonconforming finite element method¶with anisotropic mesh grading¶for the incompressible Navier–Stokes equations¶in domains with edges

Any solution of the incompressible Navier–Stokes equations in three-dimensional domains with edges has anisotropic singular behaviour which is treated numerically by using anisotropic finite element meshes. The velocity is approximated by Crouzeix–Raviart (nonconforming 𝒫₁) elements and the pressure by piecewise constants. This method is stable for general meshes since the inf-sup condition is satisfied without minimal or maximal angle condition. The existence of solutions to the discrete problems follows. Consistency error estimates for the divergence equation are obtained for anisotropic tensor product meshes. As applications, the consistency error estimate for the Navier–Stokes solution and some discrete Sobolev inequalities are derived on such meshes. These last results provide optimal error estimates in the uniqueness case by the use of appropriately refined anisotropic tensor product meshes, namely, if N _e is the number of elements of the mesh, we prove that the optimal order of convergence h∼ N _e ^{− 1/3}. Received:July 2001 / Accepted: July 2002     

On the Attainable Order of Collocation Methods for the Neutral Functional-Differential Equations with Proportional Delays

In this paper, we extend the recent results of H. Brunner in BIT (1997) for the DDE y′(t)= by(qt), y(0)=1 and the DVIE y(t)=1+∫₀ ^t by(qs)ds with proportional delay qt, 0<q≤1, to the neutral functional-differential equation (NFDE): and the delay Volterra integro-differential equation (DVIDE) : with proportional delays p _i t and q _i t, 0<p _i ,q _i ≤1 and complex numbers a,b _i and c _i . We analyze the attainable order of m-stage implicit (collocation-based) Runge-Kutta methods at the first mesh point t=h for the collocation solution v(t) of the NFDE and the `iterated collocation solution u _it (t)' of the DVIDE to the solution y(t), and investigate the existence of the collocation polynomials M _m (t) of v(th) or M^ _m (t) of u _it (th), t∈[0,1] such that the rational approximant v(h) or u _it (h) is the (m,m)-Padé approximant to y(h) and satisfies |v(h)−y(h)|=O(h ^{2 m +1}). If they exist, then we actually give the conditions of M _m (t) and M^ _m (t), respectively. Received September 17, 1998; revised September 30, 1999     

Discretization and error estimates for boundary value problems of the second order

Summary Boundary value problems for the equationLy≡−[p(x)y′]′+q(x)y=f(x) are considered. The method of finite differences is applied in a usual way andO (h ²) estimates are given for the discretization error as well as for its first and second difference quotients.     

Error Analysis for hp-FEM Semi-Lagrangian Second Order BDF Method for Convection-Dominated Diffusion Problems

We present in this paper an analysis of a semi-Lagrangian second order Backward Difference Formula combined with hp-finite element method to calculate the numerical solution of convection diffusion equations in ℝ². Using mesh dependent norms, we prove that the a priori error estimate has two components: one corresponds to the approximation of the exact solution along the characteristic curves, which is O(Dt²+h^m+1(1+\frac\mathopen|logh|Dt))O(\Delta t^{2}+h^{m+1}(1+\frac{\mathopen{|}\log h|}{\Delta t})); and the second, which is O(Dt^p+|| [(u)\vec]-[(u)\vec]_h||_L^￥)O(\Delta t^{p}+\| \vec{u}-\vec{u}_{h}\|_{L^{\infty}}), represents the error committed in the calculation of the characteristic curves. Here, m is the degree of the polynomials in the finite element space, [(u)\vec]\vec{u} is the velocity vector, [(u)\vec]_h\vec{u}_{h} is the finite element approximation of [(u)\vec]\vec{u} and p denotes the order of the method employed to calculate the characteristics curves. Numerical examples support the validity of our estimates.     

Analysis of Galerkin Methods for the Fully Nonlinear Monge-Ampère Equation

This paper develops and analyzes finite element Galerkin and spectral Galerkin methods for approximating viscosity solutions of the fully nonlinear Monge-Ampère equation det (D ² u ⁰)=f (>0) based on the vanishing moment method which was developed by the authors in Feng and Neilan (J. Sci. Comput. 38:74–98, 2009) and Feng (Convergence of the vanishing moment method for the Monge-Ampère equation, submitted). In this approach, the Monge-Ampère equation is approximated by the fourth order quasilinear equation −εΔ² u ^ε +det D ² u ^ε =f accompanied by appropriate boundary conditions. This new approach enables us to construct convergent Galerkin numerical methods for the fully nonlinear Monge-Ampère equation (and other fully nonlinear second order partial differential equations), a task which has been impracticable before. In this paper, we first develop some finite element and spectral Galerkin methods for approximating the solution u ^ε of the regularized problem. We then derive optimal order error estimates for the proposed numerical methods. In particular, we track explicitly the dependence of the error bounds on the parameter ε, for the error u^e-u^e_hu^{\varepsilon}-u^{\varepsilon}_{h}. Due to the strong nonlinearity of the underlying equation, the standard error estimate technique, which has been widely used for error analysis of finite element approximations of nonlinear problems, does not work here. To overcome the difficulty, we employ a fixed point technique which strongly makes use of the stability property of the linearized problem and its finite element approximations. Finally, using the Argyris finite element method as an example, we present a detailed numerical study of the rates of convergence in terms of powers of ε for the error u⁰-u_h^eu^{0}-u_{h}^{\varepsilon}, and numerically examine what is the “best” mesh size h in relation to ε in order to achieve these rates.     

Finite Element Approximation of a Three Dimensional Phase Field Model for Void Electromigration

We consider a finite element approximation of a phase field model for the evolution of voids by surface diffusion in an electrically conducting solid. The phase field equations are given by the nonlinear degenerate parabolic system

Periods in Extensions of Words

Let π(w) denote the minimum period of the word w,let w be a primitive word with period π(w) < |w|, and let z be a prefix of w. It is shown that if π(wz) = π(w), then |z| < π(w) − gcd (|w|, |z|). Detailed improvements of this result are also proven. Finally, we show that each primitive word w has a conjugate w′ = vu, where w = uv, such that π(w′) = |w′| and |u| < π(w). As a corollary we give a short proof of the fact that if u,v,w are words such that u ² is a prefix of v ², and v ² is a prefix of w ², and v is primitive, then |w| > 2|u|.     

Quadrature based finite element approximations to time dependent parabolic equations with nonsmooth initial data

The purpose of this paper is to study the effect of numerical quadrature in the finite element analysis for a time dependent parabolic equation with nonsmooth initial data. Both semidiscrete and fully discrete schemes are analyzed using standard energy techniques. For the semidiscrete case, optimal order error estimates are derived in the L ² and H ¹-norms and quasi-optimal order in the L ^∞-norm, when the initial function is only in H ₀ ¹. Finally, based on the backward Euler method, a time discretization scheme is discussed and almost optimal rates of convergence in the L ², H ¹ and L ^∞-norms are established. Received: September 1997 / Accepted: October 1997     

Teoremi di confronto per gli zeri di polinomis-ortogonali

Sommaire Dans cette note on donne des théorèmes généraux qui permettent la comparaison entre les zéros des polyn?mess-orthogonaux, relatifs à fonctions poidsp _x (x),p ₀ (x), pour les quelles le rapportp ₁ (x)/p _o(x) est une fonction strictement monotone sur un certain intervalle. Ces thèorèmes, utilisés à propos, permettent d'établir des inégalités entre les zéros des polynomess-orthogonaux relatifs aux fonctions poids de type Jacobi,p(x)=(1-x)^α (1+x)^β), α, β>−1,x∈]−1,1[; inégalités qui sont connues seulement pour des valeurs particuliers de α,β,s, et du degrém du polynome.     

Cyclic codes over R = Fp + uFp +?+ uFp with length pn

In this paper, all cyclic codes with length p^sn, (n prime to p) over the ring R = F_p + uF_p +?+ u^k−1F_p are classified. It is first proved that Tor_j(C) is an ideal of , so that the structure of ideals over extension ring S_u^k(m,ω)=GR(u^k,m)[ω]/〈ω^ps-1〉 is determined. Then, an isomorphism between R[X]/〈X^N − 1〉 and a direct sum ⊕_h∈IS_u^k(m_h,ω) can be obtained using discrete Fourier transform. The generator polynomial representation of the corresponding ideals over F_p + uF_p +?+ u^k−1F_p is calculated via the inverse isomorphism. Moreover, torsion codes, MS polynomial and inversion formula are described.