Convergence of goal‐oriented adaptive finite element methods for nonsymmetric problems

In this article, we develop convergence theory for a class of goal‐oriented adaptive finite element algorithms for second‐order nonsymmetric linear elliptic equations. In particular, we establish contraction results for a method of this type for Dirichlet problems involving the elliptic operator with A Lipschitz, symmetric positive definite, with b divergence‐free, and with . We first describe the problem class and review some standard facts concerning conforming finite element discretization and error‐estimate‐driven adaptive finite element methods (AFEM). We then describe a goal‐oriented variation of standard AFEM. Following the recent work of Mommer and Stevenson for symmetric problems, we establish contraction and convergence of the goal‐oriented method in the sense of the goal function. Our analysis approach is signficantly different from that of Mommer and Stevenson, combining the recent contraction frameworks developed by Cascon, Kreuzer, Nochetto, and Siebert; by Nochetto, Siebert, and Veeser; and by Holst, Tsogtgerel, and Zhu. We include numerical results, demonstrating performance of our method with standard goal‐oriented strategies on a convection problem. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 479–509, 2016     

Remarks on sufficient conditions for fixed points of multivalued nonexpansive mappings

In recent papers [S. Dhompongsa, T. Domínguez-Benavides, A. Kaewcharoen, A. Kaewkhao, B. Panyanak, The Jordan–von Neumann constants and fixed points for multivalued nonexpansive mappings, J. Math. Anal. Appl. 320 (2006) 916–927; S. Dhompongsa, A. Kaewcharoen, A. Kaewkhao, The Domínguez–Lorenzo condition and multivalued nonexpansive mappings, Nonlinear Anal. 64 (2006) 958–970], two sufficient conditions, namely the Domínguez–Lorenzo condition and property (D$D$), for fixed points of multivalued nonexpansive mappings are introduced. The authors also give some sufficient conditions for the Domínguez–Lorenzo condition and property (D$D$). In their proofs, it seems reasonable to use the James and von Neumann–Jordan constants. With slight modifications, significant improvements are obtained. Some new estimates are tight in Hilbert spaces.     

关于圆簿板大挠度问题的正交条件解法

本文重新考察了钱伟长教授求解圆薄板大挠度问题的系统近似法,发现此法实质上可视为奇异摄动理论中的变形参数法.以无量纲中心挠度为小参数,将挠度、中面薄膜力和载荷参数作渐近展开,我们对所得的递推方程给出了正交条件(可解性条件),据此可确定圆薄板的刚度特性.本文指出,利用圆薄板小挠度解和正交条件,可以不经求解方程而导得载荷参数与中心挠度关系的三阶近似以及中心点、边缘处的薄膜力的首项近似.文中对若干特例(均布载荷、复合载荷、各种边界条件)进行了具体计算,所得的结果与钱伟长、叶开沅、黄黔等人在文[1～4]中给出的结果完全相符.     

An implicit solution of bi-penalty approximation with orthogonality projection for the numerical simulation of Bingham fluid flow

1.IntroductionItisamajordiffct.encefi-omtheNewtonnuidflowthattheBinghammodelofNonNewtonfluidflowischaracterizedbytwoparameters:ayieldstressandaviscosity.WhenthestressoftheBinghalnmaterialbelowtheyieldstress,materialisrigidotherwisethequasiNewtolliannowresultstll:'71.Hence,therearesomeofthefloating"rigidcores"involvedintheBinghamfluidfloworsomeofthe'rigidcores"attachedtotheboundaries,inwhichthelocationsalldshapesofthese"rigidcores"maychangeforthetransientBinghamfluid,flow.ThisBingllammodelh…     

三角形单元协调与非协调位移的能量正交关系

基于组合稳定化变分原理,周天孝提出的组合杂交法是绝对收敛和稳定的,它给出了一种系统化的增强应力/应变方法,并建立了一簇低阶仿射等价的n-cube(n=2,3)单元。本文论证了单元上应力插值为线性,位移插值为协调线性部分和非协调二次部分之和的三角形组合杂交单元其协调部分与非协调部分的能量正交关系,进而得到此三角形单元刚度矩阵等同于协调的三角形线性元刚度矩阵,即非协调部分无应变增强特性。     

The energy orthogonal relation between conforming and non-conforming displacements of triangular element

Based on the variational principle of combinative stability, combined hybrid methods posed by Zhou Tianxiao are absolutely convergent and stabilized. Zhou has advocated a systematic approach to enhanced stress/strain schemes and has designed a family of lower-order elements which are affine-equivalent ton-cube (n=2,3). The energy orthogonal relation between the conforming part and the non-conforming part of displacements interpolation functions in triangular element is given, in which the stress is interpolated by linear polynomials on each element, but the displacements are interpolated by the sum of conforming linear and non-conforming quadratic polynomials. Furthermore, this element is equivalent to the conforming triangular linear element, that is, the non-conforming parts have no contribution to enhanced strains.     

Multipoint Padé approximants to complex Cauchy transforms with polar singularities

We study diagonal multipoint Padé approximants to functions of the form

where R is a rational function and λ is a complex measure with compact regular support included in , whose argument has bounded variation on the support. Assuming that interpolation sets are such that their normalized counting measures converge sufficiently fast in the weak-star sense to some conjugate-symmetric distribution σ, we show that the counting measures of poles of the approximants converge to , the balayage of σ onto the support of λ, in the weak^* sense, that the approximants themselves converge in capacity to F outside the support of λ, and that the poles of R attract at least as many poles of the approximants as their multiplicity and not much more.     

Pseudo-localization of singular integrals and noncommutative Calderón-Zygmund theory

The weak type (1,1) boundedness of singular integrals acting on matrix-valued functions has remained open since the 1980s, mainly because the methods provided by the vector-valued theory are not strong enough. In fact, we can also consider the action of generalized Calderón-Zygmund operators on functions taking values in any other von Neumann algebra. Our main tools for its solution are two. First, the lack of some classical inequalities in the noncommutative setting forces to have a deeper knowledge of how fast a singular integral decreases—L₂ sense—outside of the support of the function on which it acts. This gives rise to a pseudo-localization principle which is of independent interest, even in the classical theory. Second, we construct a noncommutative form of Calderón-Zygmund decomposition by means of the recent theory of noncommutative martingales. This is a corner stone in the theory. As application, we obtain the sharp asymptotic behavior of the constants for the strong Lp inequalities, also unknown up to now. Our methods settle some basics for a systematic study of a noncommutative Calderón-Zygmund theory.     

Generalized Hadamard matrices whose transposes are not generalized Hadamard matrices

In answer to “Research Problem 16” in Horadam's recent book Hadamard matrices and their applications, we provide a construction for generalized Hadamard matrices whose transposes are not generalized Hadamard matrices. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 456–458, 2009     

杂交元零能模式抑制的正交基本变形模式方法

通过杂交元位移场直接推导了单元基本变形模式,并且采用联合正交条件提出一种新的正交化方法,所得到的正交基本变形模式不仅具有简单的变形特征而且和材料参数无关,可以方便有效地考察单元性能,为评价不同杂交元提供了一个统一的参考标准．在此基础上利用柔度矩阵正定性给出一种简单有效的零能模式识别方法,并进一步利用基本变形模式和初始应力模式之间耦合关系,提出一种抑制杂交元零能模式的假设应力场方法,同时指出基本变形模式正交性是抑制单元零能模式的充分必要条件．2D-4节点和3D-8节点单元的数值算例说明了该文基本变形模式方法的有效性．