Weak Galerkin finite element methods with or without stabilizers

The purpose of this paper is to investigate the connections between the weak Galerkin (WG) methods with and without stabilizers. The choices of stabilizers directly affect the convergence rates of the corresponding WG methods in general. However, we observed that the convergence rates are independent of the choices of stabilizers for these WG elements with stabilizers being optional. In this paper, we will verify such phenomena theoretically as well as numerically.

     

Conforming and nonconforming harmonic finite elements

ABSTRACT

Instead of using the full polynomial space, a conforming and a nonconforming finite element methods are designed where only harmonic polynomials (a much smaller space) are employed in the computation. The conforming quadratic harmonic polynomial finite element is defined only on a special triangular grid. The nonconforming quadratic harmonic finite element is defined on general triangular grids. The optimal order of convergence is proved for both finite element methods, and confirmed by numerical computations. In addition, numerical comparisons with the standard conforming and nonconforming finite elements are presented.     

Implementing and using high-order finite element methods

We present theoretical analyses of and detailed timings for two programs which use high-order finite element methods to solve the Navier- Strokes equations in two and three dimensions. The analyses show that algorithms popular in low-order finite element implementations are not always appropriate for high-order methods. The timings show that with the proper algorithms high-order finite element methods are viable for solving the Navier-Stokes equations. We show that it is more efficient, both in time and storage, not to precompute element matrices as the degree of approximating functions increases. We also study the cost of assembling the stiffness matrix versus directly evaluating bilinear forms in two and three dimensions. We show that it is more efficient not to assemble the full stiffness matrix for high-order methods in some cases. We consider the computational issues with regard to both Euclidean and isoparametric elements. We show that isoparametric elements may be used with higher-order elements without increasing the order of computational complexity.     

Treatments of discontinuity and bubble functions in the multigrid method

When multilevel finite element spaces are not nested, different intergrid transfer operators would lead to different multigrid algorithms. It is proposed in this paper that discontinuous functions be averaged to continuous functions and that the bubble functions be discarded in the multigrid transferring. Applications of the techniques to various problems are presented with convergence analysis. Numerical comparisons with other existing methods are provided.

     

AN ERROR ANALYSIS METHOD SPP-BEAM AND A CONSTRUCTION GUIDELINE OF NONCONFORMING FINITE ELEMENTS FOR FOURTH ORDER ELLIPTIC PROBLEMS

Under two hypotheses of nonconforming finite elements of fourth order elliptic problems,we present a side-patchwise projection based error analysis method(SPP-BEAM for short).Such a method is able to avoid both the regularity condition of exact solutions in the classical error analysis method and the complicated bubble function technique in the recent medius error analysis method.In addition,it is universal enough to admit generalizations.Then,we propose a sufficient condition for these hypotheses by imposing a set of in some sense necessary degrees of freedom of the shape function spaces.As an application,we use the theory to design a P3 second order triangular H2 non-conforming element by enriching two P4 bubble functions and,another P4 second order triangular H2 nonconforming finite element,and a P3 second order tetrahedral H2 non-conforming element by enriching eight P4 bubble functions,adding some more degrees of freedom.     

邻苯二胺与醛反应合成苯并咪唑衍生物的反应机理研究

以4-甲基苯甲醛与邻苯二胺反应合成2-(4-甲基苯基)苯并咪唑的反应为例,以HPLC方法跟踪反应进程,对关键中间体与产物的结构通过1H NMR,FT-IR及LC-MS等手段进行了表征.实验证明了在邻苯二胺和醛物质的量比为1:1时同时生成单席夫碱和双席夫碱中间体的反应机理,并提出了该类反应的影响规律.     

A new approach for solving stokes systems arising from a distributive relaxation method

The distributed relaxation method for the Stokes problem has been advertised as an adequate change of variables that leads to a lower triangular system with Laplace operators on the main diagonal for which multigrid methods are very efficient. We show that under high regularity of the Laplacian, the transformed system admits almost block‐lower triangular form. We analyze the distributed relaxation method and compare it with other iterative methods for solving the Stokes system. We also present numerical experiments illustrating the effectiveness of the transformation which is well established for certain finite difference discretizations of Stokes problems. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 898–914, 2011     

西布曲明对映体的非手性柱高效液相色谱法拆分

采用非手性的C18色谱柱,通过在流动相中加入手性选择剂β-环糊精的方法实现西布曲明对映体的拆分。流动相组成为含β-环糊精的甲醇-水（10：90,V／V,pH3．6）。当手性选择剂β-环糊精的浓度为0．8mmol／L时,西布曲明对映体在进样后的5min内得到了基线分离,分离度Rs达到3．2。对流动相中加入手性选择剂进行手性拆分的机理进行了初步的探讨。     

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.