一类拟线性退缩抛物型方程整体解的存在性与衰减估计

本文讨论了一类拟线性抛物型方程初边值问题整体解的存在性和衰减估计.所得结果改进并推广了文献[1]的相应结果.     

Remarks on Sharp Interface Limit for an Incompressible Navier-Stokes and Allen-Cahn Coupled System*

The authors are concerned with the sharp interface limit for an incompressible Navier-Stokes and Allen-Cahn coupled system in this paper. When the thickness of the diffuse interfacial zone, which is parameterized by ε, goes to zero, they prove that a solution of the incompressible Navier-Stokes and Allen-Cahn coupled system converges to a solution of a sharp interface model in the L^∞(L²) ∩ L²(H¹) sense on a uniform time interval independent of the smal...     

A Low Order Nonconforming Mixed Finite Element Method for Non-Stationary Incompressible Magnetohydrodynamics System

A low order nonconforming mixed finite element method (FEM) is established for the fully coupled non-stationary incompressible magnetohydrodynamics (MHD) problem in a bounded domain in 3D. The lowest order finite elements on tetrahedra or hexahedra are chosen to approximate the pressure, the velocity field and the magnetic field, in which the hydrodynamic unknowns are approximated by inf-sup stable finite element pairs and the magnetic field by $H^1(\Omega)$-conforming finite elements, respectively. The existence and uniqueness of the approximate solutions are shown. Optimal order error estimates of $L^2(H^1)$-norm for the velocity field, $L^2(L^2)$-norm for the pressure and the broken $L^2(H^1)$-norm for the magnetic field are derived.     

Local Dispersive and Strichartz Estimates for the Schrödinger Operator on the Heisenberg Group

It was proved by Bahouri et al. [9] that the Schrödinger equation on the Heisenberg group $\mathbb{H}^d,$ involving the sublaplacian, is an example of a totally non-dispersive evolution equation: for this reason global dispersive estimates cannot hold. This paper aims at establishing local dispersive estimates on $\mathbb{H}^d$ for the linear Schrödinger equation, by a refined study of the Schrödinger kernel $S_t$ on $\mathbb{H}^d.$ The sharpness of these estimates is discussed through several examples. Our approach, based on the explicit formula of the heat kernel on $\mathbb{H}^d$ derived by Gaveau [19], is achieved by combining complex analysis and Fourier-Heisenberg tools. As a by-product of our results we establish local Strichartz estimates and prove that the kernel $S_t$ concentrates on quantized horizontal hyperplanes of $\mathbb{H}^d.$     

Approximations in Sobolev spaces by prolate spheroidal wave functions

Recently, there is a growing interest in the spectral approximation by the Prolate Spheroidal Wave Functions (PSWFs) ${\psi }_{n,c},c>0$. This is due to the promising new contributions of these functions in various classical as well as emerging applications from Signal Processing, Geophysics, Numerical Analysis, etc. The PSWFs form a basis with remarkable properties not only for the space of band-limited functions with bandwidth c, but also for the Sobolev space $H^s([?1,1])$. The quality of the spectral approximation and the choice of the parameter c when approximating a function in $H^s([?1,1])$ by its truncated PSWFs series expansion, are the main issues. By considering a function $f\in H^s([?1,1])$ as the restriction to $[?1,1]$ of an almost time-limited and band-limited function, we try to give satisfactory answers to these two issues. Also, we illustrate the different results of this work by some numerical examples.     

Error estimates for space–time discontinuous Galerkin formulation based on proper orthogonal decomposition

In this study, proper orthogonal decomposition (POD) method is applied to diffusion–convection–reaction equation, which is discretized using space–time discontinuous Galerkin (dG) method. We provide estimates for POD truncation error in dG-energy norm, dG-elliptic projection, and space–time projection. Using these new estimates, we analyze the error between the dG and the POD solution, and the error between the exact and the POD solution. Numerical results, which are consistent with theoretical convergence rates, are presented.     

双调和算子特征值问题的混合三角谱元方法

本文针对双调和算子特征值问题设计了基于混合变分形式的三角谱元逼近格式,其基函数采用指标为(-1,-1,-1)的广义Koornwinder多项式.在H~1-及H_0~1-正交谱元投影的逼近理论基础上,我们建立了双调和算子特征值与特征函数的收敛性估计;它关于网格尺寸h是最优的,关于多项式次数M是次优的.然而,在H_0~2-正交谱元投影的最优估计假设前提下,关于M的次优收敛阶估计则提升为最优.此外,Koornwinder分片多项式逼近的结果还表明,在带权Besov空间范数的度量下,对于存在着区域角点奇性的双调和算子特征值问题,谱元方法的收敛阶能达到h-型有限元方法的2倍.最后,本文的数值实验结果展示了谱元逼近格式的高效性,同时也验证了相关理论的正确性.     

Decoupled schemes for unsteady MHD equations. I. time discretization

In this article, a time discretization decoupled scheme for two‐dimensional magnetohydrodynamics equations is proposed. The almost unconditional stability and convergence of this scheme are provided. The optimal error estimates for velocity and magnet are provided, and the optimal error estimate for pressure are deduced as well. Finite element spatial discretization and numerical implementation are considered in our article (Zhang and He, Comput Math Appl 69 (2015), 1390–1406). © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 956–973, 2017