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A full multigrid method for the Steklov eigenvalue problem

INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS 2019年第12期96卷 2371-2386页

作者： Xu, Fei Beijing Univ Technol Beijing Inst Sci & Engn Comp Beijing 100124 Peoples R China

This paper introduces a kind of parallel multigrid method for solving Steklov eigenvalue problem based on the multilevel correction method. Instead of the common costly way of directly solving the Steklov eigenvalue problem on some fine space, the new method contains some boundary value problems on a series of multilevel finite element spaces and some steps of solving Steklov eigenvalue problems on a very low dimensional space. The linear boundary value problems are solved by some multigrid iteration steps. We will prove that the computational work of this new scheme is truly optimal, the same as solving the corresponding linear boundary value problem. Besides, this multigrid scheme has a good scalability by using parallel computing technique. Some numerical experiments are presented to validate our theoretical analysis.

关键词： Steklov eigenvalue problem multigrid method multilevel correction parallel computing finite element method

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Convergence of a multigrid method for elliptic equations with highly oscillatory coefficients

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SIAM JOURNAL ON NUMERICAL ANALYSIS 1997年第6期34卷 2254-2273页

作者： Engquist, B Luo, ED CARNEGIE MELLON UNIV DEPT MATHPITTSBURGHPA 15213

Standard multigrid methods are not so effective for equations with highly oscillatory coefficients. New coarse grid operators based on homogenized operators are introduced to restore the fast convergence rate of multigrid methods. Finite difference approximations are used for the discretization of the equations. Convergence analysis is based on the homogenization theory. Proofs are given for a two-level multigrid method with the homogenized coarse grid operator for two classes of two-dimensional elliptic equations with Dirichlet boundary conditions.

关键词： elliptic equation oscillation finite difference multigrid method homogenization theory convergence

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Real-space multigrid method for linear-response quantum transport in molecular electronic devices

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IEEE TRANSACTIONS ON NANOTECHNOLOGY 2007年第2期6卷 238-244页

作者： Feng, Guogang Wijesekera, Nimal Beck, Thomas L. Univ Cincinnati Dept Chem Cincinnati OH 45221 USA

We present a self-consistent ab initio simulation method to calculate coherent quantum transport through a molecule connected to metal electrodes in the linear-response regime. Density-functional theory (DFT) is applied to the metal-molecule-metal system. The molecule and the metal electrodes are treated on the same footing as one extended molecule. The Full Approximation Scheme (FAS) nonlinear multigrid technique is used to accelerate convergence in a nonorthogonal localized orbital basis. The Landauer formula is employed to calculate the current with the transmission function obtained from a Green's function calculation. The current-voltage characteristics of a benzene-1,4-dithiolate (BDT) extended molecule are studied as an example, and our results are compared to other theoretical calculations. We also show that a recently formulated constrained-current formalism is invariant to a reversal in the imposed current. Hence, the predicted voltage drop must be zero. This suggests the theory must be modified to properly treat possible nonlinearities in the nonzero current case.

关键词： density functional theory electron transport molecular electronics multigrid method

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Adaptive multigrid method for numerical solutions of elastic wave equation

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APPLIED MATHEMATICS AND COMPUTATION 2002年第2-3期133卷 609-614页

作者： Han, B Zhou, XY Liu, JQ Harbin Inst Technol Dept Math Harbin 150001 Peoples R China

Finite-element multigrid method is applied to two-dimensional elastic wave equation in frequency domain. To avoid nonsensical increase in computational cost caused by global grid refinement, an adaptive algorithm is d... 详细信息

关键词： elastic wave equation multigrid method adaptive algorithm

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CONVERGENCE ANALYSIS OF A multigrid method FOR A NONLOCAL MODEL

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SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS 2017年第3期38卷 869-890页

作者： Chen, Minghua Deng, Weihua Lanzhou Univ Gansu Key Lab Appl Math & Complex Syst Sch Math & Stat Lanzhou 730000 Gansu Peoples R China

Recently, nonlocal models attract the wide interest of scientists. They mainly come from two applied scientific fields: peridynamics and anomalous diffusion. Even though the matrices of the algebraic equation corresponding to the nonlocal models are usually Toeplitz (denote a(0) as the principal diagonal element, a(1) as the trailing diagonal element, etc). There are still some differences for the models in these two fields. For the model of anomalous diffusion, a(0)/a(1) is uniformly bounded;most of the time, a(0)/a(1) of the model for peridynamics is unbounded as the step size h tends to zero. Based on the uniform boundedness of a(0)/a(1), the convergence of the two-grid method is well established [R. H. Chan, Q.-S. Chang, and H.-W. Sun, SIAM J. Sci. Comput., 19 (1998), pp. 516-529;H. Pang and H. Sun, J. Comput. Phys., 231 (2012), pp. 693-703;M. H. Chen, Y. T. Wang, X. Cheng, and W. H. Deng, BIT, 54 (2014), pp. 623-647]. This paper provides the detailed proof of the convergence of the two-grid method for the nonlocal model of peridynamics. Some special cases of the full multigrid and the V-cycle multigrid methods are also discussed. The numerical experiments are performed to verify the convergence.

关键词： multigrid method nonlocal model Toeplitz matrices

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Calculation method for comprehensive damping of ball bearings based on multigrid method

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INDUSTRIAL LUBRICATION AND TRIBOLOGY 2020年第7期72卷 937-945页

作者： Sun, Peng Chen, Weifang Shen, Yusu Wang, Dan Nanjing Univ Aeronaut & Astronaut Coll Mech & Elect Engn Nanjing Peoples R China Nanjing Univ Aeronaut & Astronaut Nanjing Peoples R China

Purpose As an important part of the rotor system, the damping coefficient of ball bearing has a great influence on the dynamic characteristics of the system. This study aims to propose a theoretical calculation method and an experimental test method to obtain the damping coefficient of ball bearing. Design/methodology/approach Based on Hertzian contact theory and elastohydrodynamic lubrication theory, the point contact oil film damping analysis model of ball bearing is established. The comprehensive damping calculation method considering external radial load, centrifugal force, ball spin, rotational speed and lubricating oil film is derived. The multigrid method is used to obtain the oil film pressure and thickness distribution in the contact zone. The variation trend of comprehensive damping with bearing radial load, rotational speed, oil film thickness and viscosity is analyzed. The test platform is designed and the influencing factors of damping are tested. Findings The validity of the model and reliability of the test device are verified by comparing the good consistency obtained in the work. The results show that the comprehensive damping of ball bearing increases with the increase of radial load and decreases with the increase of rotational speed. Originality/value At present, the existing bearing damping model can achieve approximate calculation of damping, but the factors considered in these models are not comprehensive enough. Besides, few studies exist regarding test platform of bearing damping, and a perfect test plan has not yet been formed. In this paper, the comprehensive damping calculation model of ball bearing is improved, and a complete experimental scheme is proposed to provide reference for the comprehensive damping theory and experimental research of bearing. Peer review The peer review history for this article is available at:

关键词： Elastohydrodynamic lubrication Modal test multigrid method Ball bearing Comprehensive damping

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Covolume-based intergrid transfer operator in P₁ nonconforming multigrid method

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APPLIED NUMERICAL MATHEMATICS 2004年第1期51卷 47-67页

作者： Kang, KS Old Dominion Univ Coll Sci Dept Math & Stat Norfolk VA 23529 USA

In this paper, we introduce an intergrid transfer operator which is based on the covolume of nodes in a P-1 nonconforming multigrid method and prove the convergence of the multigrid method with this intergrid transfer operator. This intergrid transfer operator needs fewer computations and neighborhood node values than previous operators, which is a good property for parallelization. The P-1 nonconforming multigrid method with this intergrid transfer operator is suitable for solving problems with Robin boundary conditions and nonlinear problems with bound constraints on solutions. (C) 2004 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： multigrid method covolume method nonconforming finite elements elliptic equations

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Fast numerical scheme for gradient vector flow computation using a multigrid method

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IET IMAGE PROCESSING 2007年第1期1卷 48-55页

作者： Han, X. Xu, C. Prince, J. L. CMS Inc St Louis MO 63132 USA Johns Hopkins Univ Dept Elect & Comp Engn Baltimore MD 21218 USA Siemens Corp Res Imaging & Visual Dept Princeton NJ 08540 USA

The gradient vector flow (GVF) deformable model was introduced by Xu and Prince as an effective approach to overcome the limited capture range problem of classical deformable models and their inability to progress into boundary concavities. It has found many important applications in the area of medical image processing. The simple iterative method proposed in the original work on GVF, however, is slow to converge. A new multigrid method is proposed for GVF computation on 2D and 3D images. Experimental results show that the new implementation significantly improves the computational speed by at least an order of magnitude, which facilitates the application of GVF deformable models in processing large medical images.

关键词： 3D images medical image processing Biomedical measurement and imaging Biology and medical computing Computer vision and image processing techniques Linear algebra (numerical analysis) health physics Medical and biomedical uses of fields, radiations, and radioactivity 2D images Interpolation and function approximation (numerical analysis) Patient diagnostic methods and instrumentation gradient vector flow computation Optical, image and video signal processing multigrid method gradient methods Numerical approximation and analysis

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An Extrapolation Cascadic multigrid method Combined with a Fourth-Order Compact Scheme for 3D Poisson Equation

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JOURNAL OF SCIENTIFIC COMPUTING 2017年第3期70卷 1180-1203页

作者： Pan, Kejia He, Dongdong Hu, Hongling Cent S Univ Sch Math & Stat Changsha 410083 Peoples R China Tongji Univ Sch Aerosp Engn & Appl Mech Shanghai 200092 Peoples R China Hunan Normal Univ Coll Math & Comp Sci Key Lab High Performance Comp & Stochast Informat Minist Educ China Changsha 410081 Hunan Peoples R China

Extrapolation cascadic multigrid (EXCMG) method is an efficient multigrid method which has mainly been used for solving the two-dimensional elliptic boundary value problems with linear finite element discretization in the existing literature. In this paper, we develop an EXCMG method to solve the three-dimensional Poisson equation on rectangular domains by using the compact finite difference (FD) method with unequal meshsizes in different coordinate directions. The resulting linear system from compact FD discretization is solved by the conjugate gradient (CG) method with a relative residual stopping criterion. By combining the Richardson extrapolation and tri-quartic Lagrange interpolation for the numerical solutions from two-level of grids (current and previous grids), we are able to produce an extremely accurate approximation of the actual numerical solution on the next finer grid, which can greatly reduce the number of relaxation sweeps needed. Additionally, a simple method based on the midpoint extrapolation formula is used for the fourth-order FD solutions on two-level of grids to achieve sixth-order accuracy on the entire fine grid cheaply and directly. The gradient of the numerical solution can also be easily obtained through solving a series of tridiagonal linear systems resulting from the fourth-order compact FD discretizations. Numerical results show that our EXCMG method is much more efficient than the classical V-cycle and W-cycle multigrid methods. Moreover, only few CG iterations are required on the finest grid to achieve full fourth-order accuracy in both the -norm and -norm for the solution and its gradient when the exact solution belongs to . Finally, numerical result shows that our EXCMG method is still effective when the exact solution has a lower regularity, which widens the scope of applicability of our EXCMG method.

关键词： Richardson extrapolation multigrid method Compact difference scheme Quartic interpolation Poisson equation

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An efficient semi-coarsening multigrid method for variable diffusion problems in cylindrical coordinates

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APPLIED NUMERICAL MATHEMATICS 2007年第5-7期57卷 801-810页

作者： Lai, Ming-Chih Wu, Chin-Tien Tseng, Yu-Hou Natl Chiao Tung Univ Dept Appl Math Hsinchu 300 Taiwan

In this paper, we present an efficient multigrid (MG) algorithm for solving the three-dimensional variable coefficient diffusion equation in cylindrical coordinates. The multigrid V-cycle combines a semi-coarsening in azimuthal direction with the red-black Gauss-Seidel plane (radial-axial plane) relaxation. On each plane relaxation, we further semi-coarsen the axial direction with red-black line relaxation in the radial direction. We also prove the convergence of two-level MG with plane Jacobi relaxation. Numerical results show that the present multigrid method indeed is scalable with the mesh size. (c) 2006 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： multigrid method V-cycle variable diffusion equation cylindrical coordinates

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