A posteriori error estimate of the DSD method for first-order hyperbolic equations

ＩｎｔｒｏｄｕｃｔｉｏｎＩｔ’ｓｋｎｏｗｎｔｈａｔｉｎｎｕｍｅｒｉｃａｌａｐｐｒｏｘｉｍａｔｉｏｎｏｆｆｉｒｓｔ＿ｏｒｄｅｒｈｙｐｅｒｂｏｌｉｃｅｑｕａｔｉｏｎｓ,ｔｈｅｕｓｅｏｆａｄａｐｔｉｖｅｆｉｎｉｔｅｅｌｅｍｅｎｔｍｅｔｈｏｄｓ (ｓｅｅ [1 ] )ｈａｓｂｅｅｎｅｘｐａｎｄｅｄｔｏｍａｎｙｆｉｅｌｄｓｓｕｃｈａｓｃｏｍｐｕｔａｔｉｏｎａｌｆｌｏｗｍｅｃｈａｎｉｃｓ,ｔｈｅｒｍａｌａｎａｌｙｓｅｓ,ｅｌｅｃｔｒｉｃａｌｅｎｇｉｎｅｅｒｉｎｇ ,ｅｔｃ.Ｔｈｅｈ＿ｖｅｒｓｉｏｎａｄａｐｔｉｖｅｆｉｎｉ…     

一个第二类变分不等式的有限元逼近

本短文讨论下述第二类变分不等式（见 ［２, ４］）的有限元逼近及其误差分析：其中是平面凸多边形区域的的边界, 且而 ． 诸如热量控制问题,流体通过半可透性壁的扩散问题以及简化库仑摩擦接触问题的正则化方法等均可归为上述变分不等式（１）（见［２,３］）．在文［２］中给出了上述变分不等式的有限元逼近格式,作出了收敛性分析及误差估计．本文的目的是进一步用数值积分简化上述有限元逼近格式并改进原有的估计误差． 设Ｔｈ是的拟一致三角形部分,Ｖｈ是对应的线性元空间,且使得ｖｈ＝０在上．［２］中用数值积分代替其中 Ｍｉ…     

抛物问题的拉格朗日乘子区域分解法

In this paper we consider domain decomposition methods with Lagrangian multipliers, which are applied to solving parabolic problems. We shall estimate condition numbers of the resulting interface matrices, and construct two kinds of simple preconditioners for the corresponding interface equations. It will be shown that the condition numbers of the resulting preconditioned interface matrices are almost optimal.     

New subspace minimization conjugate gradient methods based on regularization model for unconstrained optimization

In this paper, two new subspace minimization conjugate gradient methods based on p-regularization models are proposed, where a special scaled norm in p-regularization model is analyzed. Different choices of special scaled norm lead to different solutions to the p-regularized subproblem. Based on the analyses of the solutions in a two-dimensional subspace, we derive new directions satisfying the sufficient descent condition. With a modified nonmonotone line search, we establish the global convergence of the proposed methods under mild assumptions. R-linear convergence of the proposed methods is also analyzed. Numerical results show that, for the CUTEr library, the proposed methods are superior to four conjugate gradient methods, which were proposed by Hager and Zhang (SIAM J. Optim. 16(1):170–192, 2005), Dai and Kou (SIAM J. Optim. 23(1):296–320, 2013), Liu and Liu (J. Optim. Theory. Appl. 180(3):879–906, 2019) and Li et al. (Comput. Appl. Math. 38(1):2019), respectively.

     

A decoupling two-grid method for the time-dependent Poisson-Nernst-Planck equations

Numerical Algorithms - We study a two-grid strategy for decoupling the time-dependent Poisson-Nernst-Planck equations describing the mass concentration of ions and the electrostatic potential. The...     

Cauchy–Born Rule and the Stability of Crystalline Solids: Static Problems

We study the connection between atomistic and continuum models for the elastic deformation of crystalline solids at zero temperature. We prove, under certain sharp stability conditions, that the correct nonlinear elasticity model is given by the classical Cauchy–Born rule in the sense that elastically deformed states of the atomistic model are closely approximated by solutions of the continuum model with stored energy functionals obtained from the Cauchy–Born rule. The analysis is carried out for both simple and complex lattices, and for this purpose, we develop the necessary tools for performing asymptotic analysis on such lattices. Our results are sharp and they also suggest criteria for the onset of instabilities of crystalline solids.     

THE MULTI-SYMPLECTIC ALGORITHM FOR “GOOD” BOUSSINESQ EQUATION

The multi-symplectic formulations of the “Good” Boussinesq equation were considered. For the multi-symplectic formulation, a new fifteen-point difference scheme which is equivalent to the multi-symplectic Preissman integrator was derived. The numerical experiments show that the multi-symplectic scheme have excellent long-time numerical behavior. Foundation items: the Foundation for Key Laboratory of Scientific/Engineering Computing Institute of Computational Mathematics and Scientific/Engineering Computing, Chinese Academy of Sciences; the Natural Science Foundation of Huaqiao University. Biography: ZENG Wen-ping (1940-), Professor (E-mail: qmz@1sec.cc.ac.cn)     

Analysis on a superlinearly convergent augmented Lagrangian method

The augmented Lagrangian method is a classical method for solving constrained optimization.Recently,the augmented Lagrangian method attracts much attention due to its applications to sparse optimization in compressive sensing and low rank matrix optimization problems.However,most Lagrangian methods use first order information to update the Lagrange multipliers,which lead to only linear convergence.In this paper,we study an update technique based on second order information and prove that superlinear convergence can be obtained.Theoretical properties of the update formula are given and some implementation issues regarding the new update are also discussed.     

Generalized skew‐Hermitian triangular splitting iteration methods for saddle‐point linear systems

A generalized skew‐Hermitian triangular splitting iteration method is presented for solving non‐Hermitian linear systems with strong skew‐Hermitian parts. We study the convergence of the generalized skew‐Hermitian triangular splitting iteration methods for non‐Hermitian positive definite linear systems, as well as spectrum distribution of the preconditioned matrix with respect to the preconditioner induced from the generalized skew‐Hermitian triangular splitting. Then the generalized skew‐Hermitian triangular splitting iteration method is applied to non‐Hermitian positive semidefinite saddle‐point linear systems, and we prove its convergence under suitable restrictions on the iteration parameters. By specially choosing the values of the iteration parameters, we obtain a few of the existing iteration methods in the literature. Numerical results show that the generalized skew‐Hermitian triangular splitting iteration methods are effective for solving non‐Hermitian saddle‐point linear systems with strong skew‐Hermitian parts. Copyright © 2013 John Wiley & Sons, Ltd.     

A note on the residual type a posteriori error estimates for finite element eigenpairs of nonsymmetric elliptic eigenvalue problems

In this paper we study the residual type a posteriori error estimates for general elliptic (not necessarily symmetric) eigenvalue problems. We present estimates for approximations of semisimple eigenvalues and associated eigenvectors. In particular, we obtain the following new results: 1) An error representation formula which we use to reduce the analysis of the eigenvalue problem to the analysis of the associated source problem; 2) A local lower bound for the error of an approximate finite element eigenfunction in a neighborhood of a given mesh element T.