HANGING NODES IN THE UNIFYING THEORY OF A POSTERIORI FINITE ELEMENT ERROR CONTROL

A unified a posteriori error analysis has been developed in [18, 21-23] to analyze the finite element error a posteriori under a universal roof. This paper contributes to the finite element meshes with hanging nodes which are required for local mesh-refining. The twodimensional 1-irregular triangulations into triangles and parallelograms and their combinations are considered with conforming and nonconforming finite element methods named after or by Courant, Q1, Crouzeix-Raviart, Poisson, Stokes and Navier-Lamé equations Han, Rannacher-Turek, and others for the The paper provides a unified a priori and a posteriori error analysis for triangulations with hanging nodes of degree ≤ 1 which are fundamental for local mesh refinement in self-adaptive finite element discretisations.     

AN IMPROVED ERROR ANALYSIS FOR FINITE ELEMENT APPROXIMATION OF BIOLUMINESCENCE TOMOGRAPHY

This paper is concerned with an ill-posed problem which results from the area of molecular imaging and is known as BLT problem. Using Tikhonov regularization technique, a quadratic optimization problem can be formulated. We provide an improved error estimate for the finite element approximation of the regularized optimization problem. Some numerical examples are presented to demonstrate our theoretical results.     

A PRIORI AND A POSTERIORI ERROR ESTIMATES OF A WEAKLY OVER-PENALIZED INTERIOR PENALTY METHOD FOR NON-SELF-ADJOINT AND INDEFINITE PROBLEMS

In this paper, we study a weakly over-penalized interior penalty method for non-self- adjoint and indefinite problems. An optimal a priori error estimate in the energy norm is derived. In addition, we introduce a residual-based a posteriori error estimator, which is proved to be both reliable and efficient in the energy norm. Some numerical testes are presented to validate our theoretical analysis.     

VARIATIONAL DISCRETIZATION FOR OPTIMAL CONTROL GOVERNED BY CONVECTION DOMINATED DIFFUSION EQUATIONS

In this paper, we study variational discretization for the constrained optimal control problem governed by convection dominated diffusion equations, where the state equation is approximated by the edge stabilization Galerkin method. A priori error estimates are derived for the state, the adjoint state and the control. Moreover, residual type a posteriori error estimates in the L^2-norm are obtained. Finally, two numerical experiments are presented to illustrate the theoretical results.     

Global Weak Solutions for the Weakly Dissipative Camassa-Holm Equation

In this paper we prove the existence and uniqueness of global weak solutions to the weakly dissipative Camassa-Holm equation.     

一类弱耗散Camassa-Holm 方程Cauchy问题在Besov 空间解的局部适定性

利用Littlewood-Paley 理论和输运方程解的先验估计, 在Besov 空间 中证明了一类弱耗散Camassa-Holm 方程Cauchy 问题解的局部适定性, 同时给出了解的能量估计及爆破准则.     

随机耗散Camassa-Holm方程的吸引子

可以接轨道得到带白噪声的随机耗散Camassa-Holm方程的唯—解并且可以检验该解产生随机动力系统,从而证明了该随机动力系统在H₀²中存在紧的随机吸引子.     

ON THE P1 POWELL-SABIN DIVERGENCE-FREE FINITE ELEMENT FOR THE STOKES EQUATIONS

The stability of the P1-P0 mixed-element is established on general Powell-Sabin triangular grids. The piecewise linear finite element solution approximating the velocity is divergence-free pointwise for the Stokes equations. The finite element solution approximating the pressure in the Stokes equations can be obtained as a byproduct if an iterative method is adopted for solving the discrete linear system of equations. Numerical tests are presented confirming the theory on the stability and the optimal order of convergence for the P1 Powell-Sabin divergence-free finite element method.     

ON THE ORDER OF SUMMABILITY OF THE FOURIER INVERSION FORMULA

In this article we show that the order of the point value, in the sense of Lojasiewicz, of a tempered distribution and the order of summability of the pointwise Fourier inversion formula are closely related. Assuming that the order of the point values and certain order of growth at infinity are given for a tempered distribution, we estimate the order of summability of the Fourier inversion formula. For Fourier series, and in other cases, it is shown that if the distribution has a distributional point value of order k, then its Fourier series is e.v. Cesaro summable to the distributional point value of order k＋1. Conversely, we also show that if the pointwise Fourier inversion formula is e.v. Cesaro summable of order k, then the distribution is the （k ＋ 1）-th derivative of a locally integrable function, and the distribution has a distributional point value of order k ＋ 2. We also establish connections between orders of summability and local behavior for other Fourier inversion problems.     

A NUMERICAL STUDY FOR THE PERFORMANCE OF THE WENO SCHEMES BASED ON DIFFERENT NUMERICAL FLUXES FOR THE SHALLOW WATER EQUATIONS

In this paper we investigate the performance of the weighted essential non-oscillatory （WENO） methods based on different numerical fluxes, with the objective of obtaining better performance for the shallow water equations by choosing suitable numerical fluxes. We consider six numerical fluxes, i.e., Lax-Friedrichs, local Lax-Friedrichs, Engquist-Osher, Harten-Lax-van Leer, HLLC and the first-order centered fluxes, with the WENO finite volume method and TVD Runge-Kutta time discretization for the shallow water equations. The detailed numerical study is performed for both one-dimensional and two-dimensional shallow water equations by addressing the property, and resolution of discontinuities. issues of CPU cost, accuracy, non-oscillatory     

The defocusing energy-supercritical Hartree equation

In this paper,we study the global well-posedness and scattering problem for the energysupercritical Hartree equation iut+Δu.(|x|.γ.|u|2)u=0 with γ4 in dimension d γ.We prove that if the solution u is apriorily bounded in the critical Sobolev space,that is,u ∈Lt∞(I;Hxsc(Rd)) with sc:= γ/2.11,then u is global and scatters.The impetus to consider this problem stems from a series of recent works for the energy-supercritical nonlinear wave equation(NLW) and nonlinear Schrdinger equation(NLS).We utilize the strategy derived from concentration compactness ideas to show that the proof of the global well-posedness and scattering is reduced to disprove the existence of three scenarios:finite time blowup;soliton-like solution and low to high frequency cascade.Making use of the No-waste Duhamel formula,we deduce that the energy of the finite time blow-up solution is zero and so get a contradiction.Finally,we adopt the double Duhamel trick,the interaction Morawetz estimate and interpolation to kill the last two scenarios.     

GLOBAL ATTRACTOR FOR CAMASSA-HOLM TYPE EQUATIONS WITH DISSIPATIVE TERM

1IntroductionWe consider the Camassa-Holm type equations with dissipative termut?uxxt δuxxxx f(u)x=ε(2uxuxx uuxxx) g(x),x∈[0,1],t>0.(1.1)Under nonlinear boundary conditionu(0,t)=ux(1,t)=uxx(0,t)=0,t>0,(1.2)δuxxx(1,t)?εu(1,t)uxx(1,t)=?(u(1,t)),t>0,(1.     

Bubble tree convergence for the harmonic sequence of harmonic surfaces in ℂℙ n

We show that any harmonic sequence determined by a harmonic map from a compact Riemannian surface M to CP^n has a terminating holomorphic （or anti-holomorphic） map from M to CP^n, or a ＂bubble tree limit＂ consisting of a harmonic map f： M → CP^n and a tree of bubbles hλ^μ： S^2 --→ CP^n.     

A POSTERIORI ERROR ESTIMATES FOR FINITE ELEMENT APPROXIMATIONS OF THE CAHN-HILLIARD EQUATION AND THE HELE-SHAW FLOW

This paper develops a posteriori error estimates of residual type for conforming and mixed finite element approximations of the fourth order Cahn-Hilliard equation ut ＋ △（ε△Au-ε^-1f（u）） = 0. It is shown that the a posteriori error bounds depends on ε^-1 only in some low polynomial order, instead of exponential order. Using these a posteriori error estimates, we construct at2 adaptive algorithm for computing the solution of the Cahn- Hilliard equation and its sharp interface limit, the Hele-Shaw flow. Numerical experiments are presented to show the robustness and effectiveness of the new error estimators and the proposed adaptive algorithm.     

SIMPLE PROOFS OF THE CAUCHY-SCHWARTZ INEQUALITY AND THE NEGATIVE DISCRIMINANTPROPERTY IN ARCHIMEDEAN ALMOST f-ALGEBRAS

The paper presents simple proofs of the Cauchy-Schwartz inequality and the negative discriminant property in archimedean almost f-algebras^[5], based on a sequence approximation.     

ON THE ERROR ESTIMATES OF A NEW OPERATE SPLITTING METHOD FOR THE NAVIER-STOKES EQUATIONS

In this paper, a new operator splitting scheme is introduced for the numerical solution of the incompressible Navier-Stokes equations. Under some mild regularity assumptions on the PDE solution, the stability of the scheme is presented, and error estimates for the velocity and the pressure of the proposed operator splitting scheme are given.     

APPLICATIONS OF THE BERNSTEIN-DURRMEYER OPERATORS IN ESTIMATING THE NORM OF MERCER KERNEL MATRICES

The paper is related to the norm estimate of Mercer kernel matrices. The lower and upper bound estimates of Rayleigh entropy numbers for some Mercer kernel matrices on [0, 1] × [0, 1] based on the Bernstein-Durrmeyer operator kernel are obtained, with which and the approximation property of the Bernstein-Durrmeyer operator the lower and upper bounds of the Rayleigh entropy number and the l2 -norm for general Mercer kernel matrices on [0, 1] x [0, 1] are provided.     

ON THE QUASI-RANDOM CHOICE METHOD FOR THE LIOUVILLE EQUATION OF GEOMETRICAL OPTICS WITH DISCONTINUOUS WAVE SPEED

We study the quasi-random choice method （QRCM） for the Liouville equation of ge- ometrical optics with discontinuous locM wave speed. This equation arises in the phase space computation of high frequency waves through interfaces, where waves undergo partial transmissions and reflections. The numerical challenges include interface, contact discon- tinuities, and measure-valued solutions. The so-called QRCM is a random choice method based on quasi-random sampling （a deterministic alternative to random sampling）. The method not only is viscosity-free but also provides faster convergence rate. Therefore, it is appealing for the prob!em under study which is indeed a Hamiltonian flow. Our analy- sis and computational results show that the QRCM 1） is almost first-order accurate even with the aforementioned discontinuities; 2） gives sharp resolutions for all discontinuities encountered in the problem; and 3） for measure-valued solutions, does not need the level set decomposition for finite difference/volume methods with numerical viscosities.     

NONLINEAR RANK-ONE MODIFICATION OF THE SYMMETRIC EIGENVALUE PROBLEM

Nonlinear rank-one modification of the symmetric eigenvalue problem arises from eigenvibrations of mechanical structures with elastically attached loads and calculation of the propagation modes in optical fiber. In this paper, we first study the existence and uniqueness of eigenvalues, and then investigate three numerical algorithms, namely Picard iteration, nonlinear Rayleigh quotient iteration and successive linear approximation method （SLAM）. The global convergence of the SLAM is proven under some mild assumptions. Numerical examples illustrate that the SLAM is the most robust method.