Quadratic functional equations in a set of Lebesgue measure zero

Let R$\mathbb{R}$ be the set of real numbers, Y   a Banach space and f:R→Y$f:\mathbb{R}\to Y$. We prove the Hyers–Ulam stability theorem for the quadratic functional inequality

‖f(x+y)+f(x−y)−2f(x)−2f(y)‖≤?$\Vert f(x+y)+f(x-y)-2f(x)-2f(y)\Vert \le ?$

for all (x,y)∈Ω$(x,y)\in \Omega$, where Ω⊂R²$\Omega \subset {\mathbb{R}}^2$ is of Lebesgue measure 0. Using the same method we dealt with the stability of two more functional equations in a set of Lebesgue measure 0.     

MIXED FINITE ELEMENT METHODS FOR A STRONGLY NONLINEAR PARABOLIC PROBLEM

1.IntroductionFOrsecondorderellipticproblems,themixedmethodwasdescribedandanalyzedbymanyauthors[1--3]inthecaseoflinearequationsindivergenceform,aswellasin[4,5]forquasilinearornonlinearproblemsindivergenceform.JohnsonandThorn6e[6]consideredalternativeproofSofthepreviouslyknownerrorestimatesforsuchmethodsintheellipticcase.Theyalsoanalyzedthemixedfiniteelementmethodforthepaxabolicequationgivenbypt--ho~f.Garcia[7]studiedtheconvergenceofmixedfiniteelementapproximationstoquasilinearparabolicequati…     

Hyperstability of a logarithmic-type functional equation on restricted domains

Let $X$ be a real normed vector space and $f:(0,\infty )\to X$. In this paper, we prove the hyperstability of the logarithmic functional equation

$f\left(x^y\right)?yf\left(x\right)=0$

on $\Gamma$ of Lebesgue measure zero. More precisely, we prove that if $f:(0,\infty )\to X$ satisfies

$6f\left(x^y\right)?yf\left(x\right)6\le ?(x,y)$

for all $(x,y)\in {\Gamma }^{+}?\{(x,y):y>\alpha \left(x\right)\}[\mathrm{resp.}{\Gamma }^{?}?\{(x,y):0<y<\alpha \left(x\right)\}]$ of Lebesgue measure zero, where $\alpha :(0,\infty )\to (0,\infty )$ is an arbitrary given function and $?:(0,\infty )\times (0,\infty )\to [0,\infty )$ satisfies the condition $?(x,y)∕y\to 0$ as $y\to \infty$ [resp. $y\to 0$], then $f$ satisfies the functional equation $f\left(x^y\right)=yf\left(x\right)$ for all $x>0$ and$y>0$.     

GLOBAL FINITE ELEMENT NONLINEAR GALERKIN METHOD FOR THE PENALIZED NAVIER-STOKES EQUATIONS GLOBAL FINITE ELEMENT NONLINEAR GALERKIN METHOD FOR THE PENALIZED NAVIER-STOKES EQUATIONS

1. IntroductionIn the numerical simulation of the Navier-Stokes equations one encounters three seriousdifficulties in the case of large Reynolds numbers f the treatment of the incomPressibility con-dition divu = 0, the treatment of the noIilinear terms and the large time integration. For thetreatment of the incoInPressibility condition, one use the penalty method in the case of finiteelemellts [1--2l and for the treatmen of the noulinar terms and the large tfor integration, oneuse the nonlin…     

A FINITE VOLUME ELEMENT METHOD FOR THERMAL CONVECTION PROBLEMS

Consider the finite volume element method for the thermal convection problem with the infinite Prandtl number. The author uses a conforming piecewise linear function on a fine triangulation for velocity and temperature, and a piecewise constant function on a coarse triangulation for pressure. For general triangulation the optimal order H^1 norm error estimates are given.     

A FINITE DIMENSIONAL METHOD FOR SOLVING NONLINEAR ILL-POSED PROBLEMS

1.IntroductionInthispaperweconsiderthenonlinearproblemsoftheformF(x)=co,(1)whereF:D(F)CX-- YisanonlinearoperatorbetweenrealHilbertspacesXandYandcoER(F).ThenormsinXandYwillbedenotedby11'IIxand11'flyrespectively.Wearemainlyinterestedinthoseproblemsofth...     

MULTISCALE FINITE ELEMENT METHOD FOR SUBDIVIDED PERIODIC ELASTIC STRUCTURES OF COMPOSITE MATERIALS

1. IntroductionFor a kind of elastic structures of composite materials of which the geometric and physicalparameters are of some periodicity3 e.g. ) common laminated plate and shell, and fibre reinforced,particle reinforced, and woved composite materials and so forth, we can regard them as theperiodic structures with a unit cell.Generally speaking, it is extremely difficult to compute directly the above elastic structuresby using usual FEM, due to the complicated geometric configurations and …     

高阶非线性波动方程的有限差分方法

本文研究一类广泛的高阶非线性波动方程组初边值问题的有限差分格式，用离散泛函分析方法和先验估计的技巧得到了有限差分格式的收敛性。     

Generic Hahn–Banach results

Given f:X→R∪{+∞}$f:X\to \mathbb{R}\cup \{+\infty \}$ a convex and lower semi-continuous function defined on a reflexive Banach space X, and L, a closed linear manifold of X over which f takes at least a real value, the aim of this note is to prove the following Baire category result: in the Euclidean setting, the set of affine functions dominated by f on L for which there is no dominated extension to X is always of first Baire category, but this set can be as large as a residual set, provided that X is a reflexive Banach space of infinite dimension.     

TIME-DISCRETIZATION PROCEDURE FOR FINITE ELEMENT APPROXIMATION OF COMPRESSIBLE MISCIBLE DISPLACEMENT IN POROUS MEDIA

We'll consider the model of two-phase compressible miscible displacement in porous media which includes molecular diffusion and dispersion in one dimensional space. Time-discretization procedure is established and analysed. The optimal error estimate in L2 norm is proved by introducing a new interpolation operator R.     

LOCAL AND PARALLEL FINITE ELEMENT ALGORITHMS FOR THE NAVIER-STOKES PROBLEM

Based on two-grid discretizations, in this paper, some new local and parallel finiteelement algorithms are proposed and analyzed for the stationary incompressible Navier-Stokes problem. These algorithms are motivated by the observation that for a solutionto the Navier-Stokes problem, low frequency components can be approximated well by arelatively coarse grid and high frequency components can be computed on a fine grid bysome local and parallel procedure. One major technical tool for the analysis is some locala priori error estimates that are also obtained in this paper for the finite element solutionson general shape-regular grids.     

On minima of a functional of the gradient: Upper and lower solutions

This paper studies a scalar minimization problem with an integral functional of the gradient under affine boundary conditions. A new approach is proposed using a minimal and a maximal solution to the convexified problem. We prove a density result: any relaxed solution continuously depending on boundary data may be approximated uniformly by solutions of the nonconvex problem keeping continuity relative to data. We also consider solutions to the nonconvex problem having Lipschitz dependence on boundary data with the best Lipschitz constant.     

FINITE ELEMENTS WITH LOCAL PROJECTION STABILIZATION FOR INCOMPRESSIBLE FLOW PROBLEMS

In this paper we review recent developments in the analysis of finite element methods for incompressible flow problems with local projection stabilization （LPS）. These methods preserve the favourable stability and approximation properties of classical residual-based stabilization （RBS） techniques but avoid the strong coupling of velocity and pressure in the stabilization terms. LPS-methods belong to the class of symmetric stabilization techniques and may be characterized as variational multiscale methods. In this work we summarize the most important a priori estimates of this class of stabilization schemes developed in the past 6 years. We consider the Stokes equations, the Oseen linearization and the NavierStokes equations. Furthermore, we apply it to optimal control problems with linear（ized） flow problems, since the symmetry of the stabilization leads to the nice feature that the operations ＂discretize＂ and ＂optimize＂ commute.     

A FINITE ELEMENT METHOD WITH RECTANGULAR PERFECTLY MATCHED LAYERS FOR THE SCATTERING FROM CAVITIES

We develop a finite element method with rectangular perfectly matched layers （PMLs） for the wave scattering from two-dimensional cavities. The unbounded computational domain is truncated to a bounded one by using of a rectangular perfectly matched layer at the open aperture. The PML parameters such as the thickness of the layer and the fictitious medium property are determined through sharp a posteriori error estimates. Numerical experiments are carried out to illustrate the competitive behavior of the proposed method.     

分次环的有限正规扩张

本文研究了群分次环的有限正规分次扩张问题．利用经典环论方法,得到一个群分次环与其有限正规分次扩张环之间关于分次Jacobson根和分次素根的关系,同时,给出了分次情形的Cutting down定理和Lying over定理．     

SUPERCONVERGENCE ANALYSIS FOR CUBIC TRIANGULAR ELEMENT OF THE FINITE ELEMENT SUPERCONVERGENCE ANALYSIS FOR CUBIC TRIANGULAR ELEMENT OF THE FINITE ELEMENT

1. IntroductionAlthough we bed proved the superconvergence of quadratic triangular elem6ntsbefore 1985, the superconvergence reseaxch of k(k 2 3)-degree triangulax elemeds Onlyhas a few advances, e.g.,.Lin, Yan and Zhou (see [15]) Prove that the three degreeHerlnite elements possess superconvergence and Walilbin (see [5-7]) obtains a roughresult by using a fine interinr estimation, that is, the placement fUnCtion or its gradestmay have weak superconvergence in the lOcal syUUnetric points,e…