A brief survey on discontinuous Galerkin methods in computational fluid dynamics

间断Galerkin （DG）方法结合了有限元法（具有弱形式、有限维解和试验函数空间）和有限体积法（具有数值通量、非线性限制器）的优点,特别适合对流占优问题（如激波等线性和非线性波）的模拟研究,本文述评DG 方法,强调其在计算流体力学（CFD）中的应用,文中讨论了DG 方法的必要构成要素和性能特点,并介绍了该方法的一些最近研究进展,相关工作促进了DG 方法在CFD 领域的应用,     

A New Discontinuous Galerkin Method for Parabolic Equations with Discontinuous Coefficient

In this paper, a new discontinuous Galerkin method is developed for the parabolic equation with jump coefficients satisfying the continuous flow condition. Theoretical analysis shows that this method is $L^2$ stable. When the finite element space consists of interpolative polynomials of degrees $k$, the convergent rate of the semi-discrete discontinuous Galerkin scheme has an order of$\mathcal{O}(h^k)$. Numerical examples for both 1-dimensional and 2-dimensional problems demonstrate the validity of the new method.     

Multirate time stepping for accelerating explicit discontinuous Galerkin computations with application to geophysical flows

This paper presents multirate explicit time‐stepping schemes for solving partial differential equations with discontinuous Galerkin elements in the framework of Large‐scale marine flows. It addresses the variability of the local stable time steps by gathering the mesh elements in appropriate groups. The real challenge is to develop methods exhibiting mass conservation and consistency. Two multirate approaches, based on standard explicit Runge–Kutta methods, are analyzed. They are well suited and optimized for the discontinuous Galerkin framework. The significant speedups observed for the hydrodynamic application of the Great Barrier Reef confirm the theoretical expectations. Copyright © 2012 John Wiley & Sons, Ltd.     

Another version of the proximal point algorithm in a Banach space

In this paper, we study some non-traditional schemes of proximal point algorithm for nonsmooth convex functionals in a Banach space. The proximal approximations to their minimal points and/or their minimal values are considered separately for unconstrained and constrained minimization problems on convex closed sets. For the latter we use proximal point algorithms with the metric projection operators and first establish the estimates of the convergence rate with respect to functionals. We also investigate the perturbed projection proximal point algorithms and prove their stability. Some results concerning the classical proximal point method for minimization problems in a Banach space is also presented in this paper.     

一类新型具有两个独立变量的不连续函数的积分不等式

主要研究了具有两个独立变量的不连续函数(有跳跃间断点)的一类更为广泛的新型积分不等式(Wendroff型),得出了一些新的结论,从而推广了前人的工作.     

超凸度量空间中非扩张映象不动点的逼近

研究了超凸度量空间中非扩张映象不动点的逼近问题,得到了具误差的Ishikawa迭代序列收敛到不动点的一个充要条件．     

Chaotic dynamics in periodically forced asymmetric ordinary differential equations

Using a topological approach, we prove the existence of infinitely many periodic solutions and the presence of chaotic dynamics for the periodically forced second order ODE u^″+bu⁺−au⁻=p(t). The choice of the equation is motivated by the studies about the Dancer-Fu?ik spectrum and the Lazer-McKenna suspension bridge model.     

Iterative schemes for computing fixed points of nonexpansive mappings in Banach spaces

Let X be a real Banach space with a normalized duality mapping uniformly norm-to-weak^? continuous on bounded sets or a reflexive Banach space which admits a weakly continuous duality mapping JΦ with gauge ?. Let f be an α-contraction and {Tn} a sequence of nonexpansive mappings, we study the strong convergence of explicit iterative schemes

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Hybrid Steepest Descent Methods for Zeros of Nonlinear Operators with Applications to Variational Inequalities

In this paper, the hybrid steepest descent methods are extended to develop new iterative schemes for finding the zeros of bounded, demicontinuous and φ-strongly accretive mappings in uniformly smooth Banach spaces. Two iterative schemes are proposed. Strong convergence results are established and applications to variational inequalities are given. In this research, the first author was partially supported by the National Science Foundation of China (10771141), Ph.D. Program Foundation of Ministry of Education of China (20070270004), and Science and Technology Commission of Shanghai Municipality (075105118). The third author was partially supported by Grant NSC 96-2628-E-110-014-MY3.