Analysis of a Fully Discrete Finite Element Method for Parabolic Interface Problems with Nonsmooth Initial Data

This article concerns numerical approximation of a parabolic interface problem with general $L^2$ initial value. The problem is discretized by a finite element method with a quasi-uniform triangulation of the domain fitting the interface, with piecewise linear approximation to the interface. The semi-discrete finite element problem is furthermore discretized in time by the $k$-step backward difference formula with $ k=1,\ldots,6 $. To maintain high-order convergence in time for possibly nonsmooth $L^2$ initial value, we modify the standard backward difference formula at the first $k-1$ time levels by using a method recently developed for fractional evolution equations. An error bound of $\mathcal{O}(t_n^{-k}\tau^k+t_n^{-1}h^2|\log h|)$ is established for the fully discrete finite element method for general $L^2$ initial data.     

Energy Identities and Stability Analysis of the Yee Scheme for 3D Maxwell Equations

In this paper numerical energy identities of the Yee scheme on uniform grids for three dimensional Maxwell equations with periodic boundary conditions are proposed and expressed in terms of the $L^2$, $H^1$ and $H^2$ norms. The relations between the $H^1$ or $H^2$ semi-norms and the magnitudes of the curls or the second curls of the fields in the Yee scheme are derived. By the $L^2$ form of the identity it is shown that the solution fields of the Yee scheme is approximately energy conserved. By the $H^1$ or $H^2$ semi norm of the identities, it is proved that the curls or the second curls of the solution of the Yee scheme are approximately magnitude (or energy)-conserved. From these numerical energy identities, the Courant-Friedrichs-Lewy (CFL) stability condition is re-derived, and the stability of the Yee scheme in the $L^2$, $H^1$ and $H^2$ norms is then proved. Numerical experiments to compute the numerical energies and convergence orders in the $L^2$, $H^1$ and $H^2$ norms are carried out and the computational results confirm the analysis of the Yee scheme on energy conservation and stability analysis.     

A Two-Grid Finite Element Method for Time-Dependent Incompressible Navier-Stokes Equations with Non-Smooth Initial Data

We analyze here, a two-grid finite element method for the two dimensional time-dependent incompressible Navier-Stokes equations with non-smooth initial data. It involves solving the non-linear Navier-Stokes problem on a coarse grid of size $H$ and solving a Stokes problem on a fine grid of size $h, h <相似文献   

具有退化粘性的非齐次双曲守恒律方程的Cauchy问题

该文讨论了如下具有退化粘性的非齐次双曲守恒律方程的Cauchy问题$\left\{\begin{array}{l} u_t+f(u)_x=a^2t^\alpha u_{xx}+g(u),\ \ \ x\in{\bf R},\ \ \ t>0,\\u(x,0)=u_0(x) \in L^\infty({\bf R}).\end{array}\right.\eqno{({\rm I})}$其中$f(u), g(u)$是${\bf R}$上的光滑函数, $a>0, 0<\alpha<1$均为常数.在此条件下, 作者首先给出了Cauchy问题(I)的局部解的存在性, 再利用极值原理获得了解的$L^{\infty}$估计, 从而证明了Cauchy问题(I)整体光滑解的存在性.     

基于Legendre多项式的数值微分: 加权$L^2$空间中的收敛性分析

We consider the problem of estimating the derivative of a function f from its noisy version fδby using the derivatives of the partial sums of Fourier-Legendre series of f~δ. Instead of the observation L~2 space, we perform the reconstruction of the derivative in a weighted L~2 space. This takes full advantage of the properties of Legendre polynomials and results in a slight improvement on the convergence order. Finally, we provide several numerical examples to demonstrate the efficiency of the proposed method.     

$L^p(K)$中RBF神经网络的系统识别问题

L^p approximation problems in system identification with RBF neural networks are investigated. It is proved that by superpositions of some functions of one variable in L^ploc（R）, one can approximate continuous functionals defined on a compact subset of L^P（K） and continuous operators from a compact subset of L^p1 （K1） to a compact subset of L^p2 （K2）. These results show that if its activation function is in L^ploc（R） and is not an even polynomial, then this RBF neural networks can approximate the above systems with any accuracy.     

具有随机隐单元的渐增前馈神经网络的$L^2(R^d)$逼近能力

This paper studies approximation capability to L2(Rd) functions of incremental constructive feedforward neural networks(FNN) with random hidden units.Two kinds of therelayered feedforward neural networks are considered:radial basis function(RBF) neural networks and translation and dilation invariant(TDI) neural networks.In comparison with conventional methods that existence approach is mainly used in approximation theories for neural networks,we follow a constructive approach to prove that one may simply randomly choose parameters of hidden units and then adjust the weights between the hidden units and the output unit to make the neural network approximate any function in L2(Rd) to any accuracy.Our result shows given any non-zero activation function g :R+→R and g(x Rd) ∈ L2(Rd) for RBF hidden units,or any non-zero activation function g(x) ∈ L2(Rd) for TDI hidden units,the incremental network function fn with randomly generated hidden units converges to any target function in L2(Rd) with probability one as the number of hidden units n→∞,if one only properly adjusts the weights between the hidden units and output unit.     

A NEW REGULARITY CLASS FOR THE NAVIER-STOKES EQUATIONS IN IR~n

ＡＮＥＷＲＥＧＵＬＡＲＩＴＹＣＬＡＳＳＦＯＲＴＨＥＮＡＶＩＥＲ－ＳＴＯＫＥＳＥＱＵＡＴＩＯＮＳＩＮＩＲ~ｎ￥Ｈ．ＢＥＩＲａＯＤＡＶＥＩＧＡ（ＤｅｐｏｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＰｉｓａＵｎｉｖｅｒｓｉｔｙ，Ｐｉｓａ，Ｉｔａｌｙ）Ａｂｓｔｒａ?..     

三维半导体问题的迎风有限体积格式

半导体器件的瞬时状态由包含三个拟线性偏微分方程所组成的方程组的初边值问题来描述.其中电子位势方程是椭圆型的,电子和空穴浓度方程是对流扩散型的.作者对三维半导体模型问题采用四面体网格上的有限体积元方法进行逼近,具体地,对电子位势方程采用一次元有限体积法来逼近,对电子浓度和空穴浓度方程采用迎风有限体积方法来逼近,并进行了详细的理论分析,得到了O(h+\Delta t)阶的L^2模误差估计结果.     

一类非牛顿流体模型解的渐近性态(英)

本文主要讨论一类带 $p \,\,( 1+\frac{2n}{n+2} \leq p<3 )\,$ 幂增长耗散位势的非牛顿流体模型解的渐近性态, 利用改进的 Fourier分解方法, 证明了其解在$L^2$ 范数下衰减率为 $(1+t)^{-\frac{n}{4}}$.     

一般无界区域中带有阻尼的三维可压缩欧拉方程

考虑在一般的三维无界区域中的具有滑移边界条件的带有阻尼的可压缩欧拉方程.当初始值接近平衡态时,获得了全局存在性和唯一性.同时,研究了在半空间情形下系统的衰减率.证明了经典解的L~2范数以(1+t)~(-3/4)衰减到常值背景解.     

Random attractors for non-autonomous fractional stochastic Ginzburg-Landau equations on unbounded domains

This paper deals with the dynamical behavior of solutions for non-autonomous stochastic fractional Ginzburg-Landau equations driven by additive noise with $\alpha\in(0,1)$. We prove the existence and uniqueness of tempered pullback random attractors for the equations in $L^{2}(\mathbf{R}^{3})$. In addition, we also obtain the upper semicontinuity of random attractors when the intensity of noise approaches zero. The main difficulty here is the noncompactness of Sobolev embeddings on unbounded domains. To solve this, we establish the pullback asymptotic compactness of solutions in $L^{2}(\mathbf{R}^{3})$ by the tail-estimates of solutions.     

生成元一致连续的多维倒向随机微分方程的$L^p$解

在生成元~$g$~的第~$i$~个分量~$g_i(t,y,z)$~仅仅依赖于矩阵~$z$~的第~$i$ 行的条件下, Hamad\`{e}ne于2003年证明了生成元一致连续的倒向随机微分 方程的~$L^2$~解的存在性, 其~$L^2$~解的唯一性由范胜君等于2010年得到. 本文进一步地证明了该类倒向随机微分 方程的~$L^p\ (p>1)$~解的存在唯一性.     

Boundary value problems for higher order parabolic equations

We consider a constant coefficient parabolic equation of order and establish the existence of solutions to the initial-Dirichlet problem in cylindrical domains. The lateral data is taken from spaces of Whitney arrays which essentially require that the normal derivatives up to order lie in with respect to surface measure. In addition, a regularity result for the solution is obtained if the data has one more derivative. The boundary of the space domain is given by the graph of a Lipschitz function. This provides an extension of the methods of Pipher and Verchota on elliptic equations to parabolic equations.

     

WAVELET DECOMPOSITIONS IN $L^2(R^2)$

Let ${V_k}^{+\infinity}_{k=-\infinity}$ be a multiresolution analysis generated by a function $\phi(x)\in L^2(R^2)$. Under this multiresolution framework the key point for studying wavelet decompositions in $L^2(R^2)$ is to study the properties of Wo which is the orthogonal complement of $V_0$ in $V_1:V_1=V_0\oplus W_0$.In this paper the author studies the structure of W_0 and furthermore shows that a box spline of three directions can generate a wavelet decomposition of $L^2(R^2)$.     

半导体瞬态问题的修正迎风有限体积格式

半导体器件的瞬时状态由包含三个拟线性偏微分方程所组成的方程组的初边值问题来描述.其中电子位势方程足椭圆型的,电子和空穴浓度方程是对流扩散型的.对电子位势方程采用一次元有限体积法米逼近,对电子浓度和空穴浓度方程采用修正的迎风有限体积方法来逼近,并进行详细的理论分析,关于位势得到O(h Δt)阶的H1模误差估计结果,关于浓度得到O(h2 Δt)阶的L2模误差估计结果.最后,给出数值例子.     

An Energy Stable Filtered Backward Euler Scheme for the MBE Equation with Slope Selection

As a promising strategy to adjust the order in the variable-order BDF algorithm, a time filtered backward Euler scheme is investigated for the molecular beam epitaxial equation with slope selection. The temporal second-order convergence in the $L^2$ norm is established under a convergence-solvability-stability (CSS)-consistent time-step constraint. The CSS-consistent condition means that the maximum step-size limit required for convergence is of the same order to that for solvability and stability (in certain norms) as the small interface parameter $ε → 0^+.$ Similar to the backward Euler scheme, the time filtered backward Euler scheme preserves some physical properties of the original problem at the discrete levels, including the volume conservation, the energy dissipation law and $L^2$ norm boundedness. Numerical tests are included to support the theoretical results.     

A global superconvergent $L^{\infty}$-error estimate of mixed finite element methods for semilinear elliptic optimal control problems

In this paper, we discuss the superconvergence of mixed finite element methods for a semilinear elliptic control problem with an integral constraint. The state and costate are approximated by the order $k=1$ Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions. Approximation of the optimal control of the continuous optimal control problem will be constructed by a projection of the discrete adjoint state. It is proved that this approximation has convergence order $h^{2}$ in $L^{\infty}$-norm. Finally, a numerical example is given to demonstrate the theoretical results.     

Existence and Decay of Global Strong Solution to 3D Density-Dependent Boussinesq Equations with Vacuum

This paper is concerned with the initial boundary problem for the three-dimensional density-dependent Boussinesq equations with vacuum. We obtain the existence of the global strong solution under the initial density in the norm $L^{\infty}$ is small enough without any smallness condition of $u$ and $\theta$. Furthermore, the exponential decay rates of the solution and their derivatives in some norm was established. In addition, we show that the solution and their derivatives are monotonically decreasing with respect to time $t$ on $[0,T]$.     

$L^{\Phi}(I,X)$中的最佳同时逼近

Let X be a Banach space and Ф be an Orlicz function. Denote by L^Ф（I,X） the space of X-valued （I）-integrable functions on the unit interval I equipped with the Luxemburg norm. For f1,f2,... ,fm ∈ L^Ф（I,X）, a distance formula distv（f1,f2,... ,fm,L^Ф（I,G）） is presented, where G is a close subspace of X. Moreover, some existence and characterization results concerning the best simultaneous approximation of L^Ф （I, G） in L^Ф （I, X） axe given.