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cascadic multigrid method FOR THE MORTAR ELEMENT method FOR P1 NONCONFORMING ELEMENT

Journal of Computational Mathematics 2005年第4期23卷 425-440页

作者： Chun-jia Bi Dan-hui Hong Department of Mathematics Yantai University Shandong 264005 China Institute of Mathematics Fudan University Shanghai 200433 China

In this paper, we consider the cascadic multigrid method for the mortar P1 nonconforming element which is used to solve the Poisson equation and prove that the cascadic conjugate gradient method is accurate with optim... 详细信息

关键词： Mortar P1 nonconforming element cascadic multigrid method

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cascadic multigrid method FOR ISOPARAMETRIC FINITE ELEMENT WITH NUMERICAL INTEGRATION

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Journal of Computational Mathematics 2004年第1期22卷 123-136页

作者： Chun-jiaBi Li-kangLi DepartmentofMathematics YantaiUniversityYantai264005China DepartmentofMathematics FudanUniversityShanghai200433China

The purpose of this paper is to study the cascadic multigrid method for the secondorder elliptic problems with curved boundary in two-dimension which are discretized by the isoparametric finite element method with numerical integration. We show that the CCG method is accurate with optimal complexity and traditional multigrid smoother (likesymmetric Gauss-Seidel, SSOR or damped Jacobi iteration) is accurate with suboptimal complexity.

关键词： cascadic multigrid method Isoparametric element Numerical integration

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cascadic multigrid method for P₁-nonconforming quadrilateral element

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JOURNAL OF NUMERICAL MATHEMATICS 2008年第3期16卷 237-248页

作者： Wang, C. Huang, Z. Li, L. Tongji Univ Dept Math Shanghai 200092 Peoples R China Tongji Univ Chinese German Coll Shanghai 200092 Peoples R China Fudan Univ Dept Math Shanghai 200433 Peoples R China

In this paper, a cascadic multigrid method for P-1-nonconforming quadrilateral finite element approximation of second order elliptic problem is studied. A new intergrid transfer operator is constructed to connect the nonnested coarse and fine grid spaces. We show that the cascadic multigrid method is accurate with optimal complexity for conjugate gradient smoother, and suboptimal complexity for some other traditional multigrid smoothers. Finally, some numerical experiments are reported to support the theory.

关键词： cascadic multigrid method P-1-nonconforming quadrilateral element second order elliptic problem

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An efficient red-black skewed extrapolation cascadic multigrid method for two-dimensional Poisson equation

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COMPUTATIONAL & APPLIED MATHEMATICS 2024年第1期43卷 1-22页

作者： Xu, Yuan Lei, Siu-Long Sun, Hai-Wei Univ Macau Dept Math Macau Peoples R China

We present a red-black skewed extrapolation cascadic multigrid (SkECMG) method to solve the Poisson equation in two dimensions based on the modified standard and skewed five-point finite difference discretization. With the help of the extrapolation technique, we develop a new extrapolation operator. Applying this proposed extrapolation operator for the second-order finite difference solutions on the current and previous coarse grid, we can design a fourth-order initial value for the iterative solver on the next finer grid. The red-black Gauss-Seidel method is adopted as a smoother which is conducive to parallel implementation. Moreover, we discuss a new sifting method as a stopping criterion to reduce the number of smoothing iterations. The numerical experiment is conducted on the square domain to verify that our SkECMG algorithm can achieve high efficiency and keep less cost simultaneously.

关键词： Elliptic boundary value problem Modified skewed five-point formula cascadic multigrid method Extrapolation

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A subspace cascadic multigrid method for mortar elements

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COMPUTING 2002年第3期69卷 205-225页

作者： Braess, D Deuflhard, P Lipnikov, K Ruhr Univ Bochum Fac Math D-44780 Bochum Germany Konrad Zuse Zentrum Berlin D-14195 Berlin Germany Free Univ Berlin D-14195 Berlin Germany Univ Houston Dept Math Houston TX 77204 USA

A cascadic multigrid (CMG) method for elliptic problems with strong material jumps is proposed and analyzed. Non-matching grids at interfaces between subdomains are allowed and treated by mortar elements. The arising saddle point problems are solved by a subspace confined conjugate gradient method as smoother for the CMG. Details of algorithmic realization including adaptivity are elaborated. Numerical results illustrate the efficiency of the new subspace CMG algorithm.

关键词： cascadic multigrid method domain decomposition mortar elements non-matching grids material jumps

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A new extrapolation cascadic multigrid method for three dimensional elliptic boundary value problems

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JOURNAL OF COMPUTATIONAL PHYSICS 2017年 344卷 499-515页

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

In this paper, we develop a new extrapolation cascadic multigrid method, which makes it possible to solve three dimensional elliptic boundary value problems with over 100 million unknowns on a desktop computer in half a minute. First, by combining Richardson extrapolation and quadratic finite element (FE) interpolation for the numerical solutions on two-level of grids (current and previous grids), we provide a quite good initial guess for the iterative solution on the next finer grid, which is a third-order approximation to the FE solution. And the resulting large linear system from the FE discretization is then solved by the Jacobi-preconditioned conjugate gradient (JCG) method with the obtained initial guess. Additionally, instead of performing a fixed number of iterations as used in existing cascadic multigrid methods, a relative residual tolerance is introduced in the JCG solver, which enables us to obtain conveniently the numerical solution with the desired accuracy. Moreover, a simple method based on the midpoint extrapolation formula is proposed to achieve higher-order accuracy on the finest grid cheaply and directly. Test results from four examples including two smooth problems with both constant and variable coefficients, an H3-regular problem as well as an anisotropic problem are reported to show that the proposed method has much better efficiency compared to the classical V-cycle and W-cycle multigrid methods. Finally, we present the reason why our method is highly efficient for solving these elliptic problems. (C) 2017 Elsevier Inc. All rights reserved.

关键词： Richardson extrapolation cascadic multigrid method Elliptic equation Quadratic FE interpolation High efficiency

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On the convergence of a cascadic multigrid method for semilinear elliptic problem

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APPLIED MATHEMATICS AND COMPUTATION 2004年第2期159卷 407-417页

作者： Zhou, SZ Hu, HX Hunan Univ Dept Appl Math Changsha Hunan Peoples R China

A cascadic multigrid method for semilinear elliptic problem is analysed. It has been proved that the algorithm has optimal order of convergence in energy norm and quasi-optimal computational complexity under more gene... 详细信息

关键词： cascadic multigrid method semilinear elliptic problem computational complexity

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A cascadic multigrid method for a kind of semilinear elliptic problem

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NUMERICAL ALGORITHMS 2011年第2期58卷 143-162页

作者： Yu, Haixiong Zeng, Jinping Hunan Univ Coll Math & Econometr Changsha 410082 Hunan Peoples R China Dongguan Univ Technol Coll Comp Dongguan 523000 Guangdong Peoples R China

In this paper, we analyze a cascadic multigrid method for semilinear elliptic problems in which the derivative of the semilinear term is Holder continuous. We first investigate the standard finite element error estimates of this kind of problem. We then solve the corresponding discrete problems using the cascadic multigrid method. We prove that the algorithm has an optimal order of convergence in energy norm and quasi-optimal computational complexity. We also report some numerical results to support the theory.

关键词： Semilinear elliptic problem Holder continuous cascadic multigrid method

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A modulus-based cascadic multigrid method for elliptic variational inequality problems

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NUMERICAL ALGORITHMS 2022年第4期90卷 1777-1791页

作者： Wang, Yan Li, Chenliang Shanghai Inst Technol Coll Sci Shanghai 201418 Peoples R China Guilin Univ Elect Technol Guangxi Coll & Univ Key Lab Data Anal & Computat Sch Math & Comp Sci Guilin Peoples R China

In this paper, by using a modulus-based matrix splitting method as a smoother, a new modulus-based cascadic multigrid method is presented for solving elliptic variational inequality problems. Convergence of the new method is analyzed. Numerical experiments confirm the theoretical analysis and show the efficiency of the proposed method.

关键词： Variational inequality cascadic multigrid method Modulus-based matrix splitting Convergence

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An Edge-based cascadic multigrid method for H(curl) problems

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NUMERICAL ALGORITHMS 2024年 1-21页

作者： Wang, Jinxuan Pan, Kejia Wu, Xiaoxin Cent South Univ Sch Math & Stat HNP LAMA Changsha 410083 Peoples R China

An efficient extrapolation cascadic multigird (EXCMG) method is developed to solve large linear systems resulting from edge element discretizations of 3D H(curl)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H(\textbf{curl})$$\end{document} problems on rectangular meshes. By treating edge unknowns as defined on the midpoints of edges, following the similar idea of the nodal EXCMG method, we design a new prolongation operator for 3D edge-based discretizations, which is used to construct a high-order approximation to the finite element solution on the refined grid. This good initial guess greatly reduces the number of iterations required by the multigrid smoother. Furthermore, the divergence correction technique is employed to further speed up the convergence of the multigrid method. Numerical examples including problems with high-contrast discontinuous coefficients are presented to validate the effectiveness of the proposed EXCMG method. The edge-based EXCMG method is more efficient than the auxiliary-space Maxwell solver (AMS) for definite problems in the considered geometrical configuration, and it can also efficiently solve large-scale indefinite problems encountered in engineering and scientific fields.

关键词： Maxwell's equations cascadic multigrid method Extrapolation Divergence correction Edge element

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