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Dive into the research topics where Aihui Zhou is active.

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Featured researches published by Aihui Zhou.


Mathematics of Computation | 2001

A two-grid discretization scheme for eigenvalue problems

Jinchao Xu; Aihui Zhou

A two-grid discretization scheme is proposed for solving eigenvalue problems, including both partial differential equations and integral equations. With this new scheme, the solution of an eigenvalue problem on a fine grid is reduced to the solution of an eigenvalue problem on a much coarser grid, and the solution of a linear algebraic system on the fine grid and the resulting solution still maintains an asymptotically optimal accuracy.


Mathematics of Computation | 2000

Local and parallel finite element algorithms based on two-grid discretizations

Jinchao Xu; Aihui Zhou

A number of new local and parallel discretization and adaptive nite element algorithms are proposed and analyzed in this paper for elliptic boundary value problems. These algorithms are motivated by the observation that, for a solution to some elliptic problems, low frequency components can be approximated well by a relatively coarse grid and high frequency components can be computed on a ne grid by some local and parallel procedure. The theoretical tools for analyzing these methods are some local a priori and a posteriori estimates that are also obtained in this paper for nite element solutions on general shape-regular grids. Some numerical experiments are also presented to support the theory. In this paper, we will propose some new parallel techniques for nite element computation. These techniques are based on our understanding of the local and global properties of a nite element solution to some elliptic problems. Simply speaking, the global behavior of a solution is mostly governed by low frequency components while the local behavior is mostly governed by high frequency compo- nents. The main idea of our new algorithms is to use a coarse grid to approximate the low frequencies and then to use a ne grid to correct the resulted residue (which contains mostly high frequencies) by some local/parallel procedures. Let us now give a somewhat more detailed but informal (and hopefully infor- mative) description of the main ideas and results in this paper. We consider the following very simple model problem posed on a convex polygonal domain R 2 : ( u + bru = f; in ;


Numerische Mathematik | 2008

Convergence and optimal complexity of adaptive finite element eigenvalue computations

Xiaoying Dai; Jinchao Xu; Aihui Zhou

In this paper, an adaptive finite element method for elliptic eigenvalue problems is studied. Both uniform convergence and optimal complexity of the adaptive finite element eigenvalue approximation are proved. The analysis is based on a certain relationship between the finite element eigenvalue approximation and the associated finite element boundary value approximation which is also established in the paper.


Applied Mathematics Letters | 2007

Eigenvalues of rank-one updated matrices with some applications

Jiu Ding; Aihui Zhou

We prove a spectral perturbation theorem for rank-one updated matrices of special structure. Two applications of the result are given to illustrate the usefulness of the theorem. One is for the spectrum of the Google matrix and the other is for the algebraic simplicity of the maximal eigenvalue of a positive matrix.


Numerische Mathematik | 2008

Local and parallel finite element algorithms for the stokes problem

Yinnian He; Jinchao Xu; Aihui Zhou; Jian Li

Based on two-grid discretizations, some local and parallel finite element algorithms for the Stokes problem are proposed and analyzed in this paper. These algorithms are motivated by the observation that for a solution to the Stokes problem, low frequency components can be approximated well by a relatively coarse grid and high frequency components can be computed on a fine grid by some local and parallel procedure. One technical tool for the analysis is some local a priori estimates that are also obtained in this paper for the finite element solutions on general shape-regular grids.


Advances in Computational Mathematics | 2001

Local and Parallel Finite Element Algorithms Based on Two-Grid Discretizations for Nonlinear Problems

Jinchao Xu; Aihui Zhou

In this paper, some local and parallel discretizations and adaptive finite element algorithms are proposed and analyzed for nonlinear elliptic boundary value problems in both two and three dimensions. The main technique is to use a standard finite element discretization on a coarse grid to approximate low frequencies and then to apply some linearized discretization on a fine grid to correct the resulted residual (which contains mostly high frequencies) by some local/parallel procedures. The theoretical tools for analyzing these methods are some local a priori and a posteriori error estimates for finite element solutions on general shape-regular grids that are also obtained in this paper.


Computer Methods in Applied Mechanics and Engineering | 2001

Gradient recovery type a posteriori error estimates for finite element approximations on irregular meshes

Ningning Yan; Aihui Zhou

In this paper, the gradient recovery type a posteriori error estimators for finite element approximations are proposed for irregular meshes. Both the global and the local a posteriori error estimates are derived. Moreover, it is shown that the a posteriori error estimates is asymptotically exact on where the mesh is regular enough and the exact solution is smooth.


Physica D: Nonlinear Phenomena | 1996

Finite approximations of Frobenius-Perron operators. A solution of Ulam's conjecture to multi-dimensional transformations

Jiu Ding; Aihui Zhou

Abstract We prove that Ulams piecewise constant approximation algorithm is convergent for computing an absolutely continuous invariant measure associated with a piecewise C2 expanding transformation or a Jablonski transformation S: [0, 1]N ⊂ RN → [0, 1]N. This solves an extension of Ulams conjecture to multi-dimensions and generalizes the convergence result given by T.-Y. Li for one-dimensional transformations.


Physical Review B | 2014

Accuracy of generalized gradient approximation functionals for density-functional perturbation theory calculations

Lianhua He; Fang Liu; Geoffroy Hautier; Micael J. T. Oliveira; Miguel A. L. Marques; Fernando D. Vila; J.J. Rehr; Gian-Marco Rignanese; Aihui Zhou

We assess the validity of various exchange-correlation functionals for computing the structural, vibrational, dielectric, and thermodynamical properties of materials in the framework of density-functional perturbation theory (DFPT). We consider five generalized-gradient approximation (GGA) functionals (PBE, PBEsol, WC, AM05, and HTBS) as well as the local density approximation (LDA) functional. We investigate a wide variety of materials including a semiconductor (silicon), a metal (copper), and various insulators (SiO2 α-quartz and stishovite, ZrSiO4 zircon, and MgO periclase). For the structural properties, we find that PBEsol and WC are the closest to the experiments and AM05 performs only slightly worse. All three functionals actually improve over LDA and PBE in contrast with HTBS, which is shown to fail dramatically for α-quartz. For the vibrational and thermodynamical properties, LDA performs surprisingly very well. In the majority of the test cases, it outperforms PBE significantly and also the WC, PBEsol and AM05 functionals though by a smaller margin (and to the detriment of structural parameters). On the other hand, HTBS performs also poorly for vibrational quantities. For the dielectric properties, none of the functionals can be put forward. They all (i) fail to reproduce the electronic dielectric constant due to the well-known band gap problem and (ii) tend to overestimate the oscillator strengths (and hence the static dielectric constant).


Advances in Computational Mathematics | 2002

A Note on the Optimal L2-Estimate of the Finite Volume Element Method

Zhongying Chen; Ronghua Li; Aihui Zhou

In this note, the optimal L2-error estimate of the finite volume element method (FVE) for elliptic boundary value problem is discussed. It is shown that ‖u−uh‖0≤Ch2|ln h|1/2‖f‖1,1 and ‖u−uh‖0≤Ch2‖f‖1,p, p>1, where u is the solution of the variational problem of the second order elliptic partial differential equation, uh is the solution of the FVE scheme for solving the problem, and f is the given function in the right-hand side of the equation.

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Jiu Ding

University of Southern Mississippi

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Xiaoying Dai

Chinese Academy of Sciences

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Lianhua He

Chinese Academy of Sciences

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Huajie Chen

Chinese Academy of Sciences

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Jinchao Xu

Pennsylvania State University

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Fang Liu

Central University of Finance and Economics

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Jun Fang

Chinese Academy of Sciences

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Lihua Shen

Chinese Academy of Sciences

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Gian-Marco Rignanese

Université catholique de Louvain

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