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Published: 4 June 2021
Last updated: 26 July 2024

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1

Rüde, Ulrich. "Fully Adaptive Multigrid Methods." SIAM Journal on Numerical Analysis 30, no. 1 (February 1993): 230–48. http://dx.doi.org/10.1137/0730011.

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2

Xu, Jinchao, and Ludmil Zikatanov. "Algebraic multigrid methods." Acta Numerica 26 (May 1, 2017): 591–721. http://dx.doi.org/10.1017/s0962492917000083.

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This paper provides an overview of AMG methods for solving large-scale systems of equations, such as those from discretizations of partial differential equations. AMG is often understood as the acronym of ‘algebraic multigrid’, but it can also be understood as ‘abstract multigrid’. Indeed, we demonstrate in this paper how and why an algebraic multigrid method can be better understood at a more abstract level. In the literature, there are many different algebraic multigrid methods that have been developed from different perspectives. In this paper we try to develop a unified framework and theory that can be used to derive and analyse different algebraic multigrid methods in a coherent manner. Given a smoother$R$for a matrix$A$, such as Gauss–Seidel or Jacobi, we prove that the optimal coarse space of dimension$n_{c}$is the span of the eigenvectors corresponding to the first$n_{c}$eigenvectors$\bar{R}A$(with$\bar{R}=R+R^{T}-R^{T}AR$). We also prove that this optimal coarse space can be obtained via a constrained trace-minimization problem for a matrix associated with$\bar{R}A$, and demonstrate that coarse spaces of most existing AMG methods can be viewed as approximate solutions of this trace-minimization problem. Furthermore, we provide a general approach to the construction of quasi-optimal coarse spaces, and we prove that under appropriate assumptions the resulting two-level AMG method for the underlying linear system converges uniformly with respect to the size of the problem, the coefficient variation and the anisotropy. Our theory applies to most existing multigrid methods, including the standard geometric multigrid method, classical AMG, energy-minimization AMG, unsmoothed and smoothed aggregation AMG and spectral AMGe.
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3

Kamowitz, David, and Seymour V. Parter. "On MGR$[\nu ]$ Multigrid Methods." SIAM Journal on Numerical Analysis 24, no. 2 (April 1987): 366–81. http://dx.doi.org/10.1137/0724028.

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4

Braess, D., and R. Verfürth. "Multigrid Methods for Nonconforming Finite Element Methods." SIAM Journal on Numerical Analysis 27, no. 4 (August 1990): 979–86. http://dx.doi.org/10.1137/0727056.

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5

Apel, Thomas, and Joachim Schöberl. "Multigrid Methods for Anisotropic Edge Refinement." SIAM Journal on Numerical Analysis 40, no. 5 (January 2002): 1993–2006. http://dx.doi.org/10.1137/s0036142900375414.

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6

Brenner, Susanne C. "Overcoming Corner Singularities Using Multigrid Methods." SIAM Journal on Numerical Analysis 35, no. 5 (October 1998): 1883–92. http://dx.doi.org/10.1137/s0036142996308022.

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7

Brenner, Susanne C., and Li-yeng Sung. "Multigrid Algorithms for C0 Interior Penalty Methods." SIAM Journal on Numerical Analysis 44, no. 1 (January 2006): 199–223. http://dx.doi.org/10.1137/040611835.

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8

Wan, Feifei, Yong Yin, Qin Zhang, and Xiuquan Peng. "Analysis of parallel multigrid methods in real-time fluid simulation." International Journal of Modeling, Simulation, and Scientific Computing 08, no. 04 (December 2017): 1750042. http://dx.doi.org/10.1142/s1793962317500428.

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The multigrid method has been widely used in computational fluid dynamics (CFD) numerical calculations because of its strong convergence. To achieve real-time simulation of a fluid in computer graphics (CG), the operation efficiency is also a significant factor to consider except for operational accuracy. For this problem, we introduced two multigrid cycling schemes, V-Cycle and full multigrid (FMG). Moreover, we have proposed a simple geometric multigrid method (GMG), and compared with the existing wide application of algebraic multigrid (AMG). All the calculations are the solution of parallel computing of GPU in this paper. The results showed that our approaches have improved the algorithm’s computational speed and convergence time, which prominently enhanced the efficiency of the fluid simulation.
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9

Goldstein, Charles I. "Multigrid Methods for Elliptic Problems in Unbounded Domains." SIAM Journal on Numerical Analysis 30, no. 1 (February 1993): 159–83. http://dx.doi.org/10.1137/0730008.

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10

Boal, Natalia, Francisco Jos´e Gaspar, Francisco Lisbona, and Petr Vabishchevich. "FINITE-DIFFERENCE ANALYSIS FOR THE LINEAR THERMOPOROELASTICITY PROBLEM AND ITS NUMERICAL RESOLUTION BY MULTIGRID METHODS." Mathematical Modelling and Analysis 17, no. 2 (April 1, 2012): 227–44. http://dx.doi.org/10.3846/13926292.2012.662177.

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This paper deals with the numerical solution of a two-dimensional thermoporoelasticity problem using a finite-difference scheme. Two issues are discussed: stability and convergence in discrete energy norms of the finite-difference scheme are proved, and secondly, a distributive smoother is examined in order to find a robust and efficient multigrid solver for the corresponding system of equations. Numerical experiments confirm the convergence properties of the proposed scheme, as well as fast multigrid convergence.
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11

Notay, Yvan. "Convergence Analysis of Perturbed Two‐Grid and Multigrid Methods." SIAM Journal on Numerical Analysis 45, no. 3 (January 2007): 1035–44. http://dx.doi.org/10.1137/060652312.

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12

El Houda, Nesba Nour, Beggas Mohammed, Belouafi Essaid, Imtiaz Ahmad, Hijaz Ahmad, and Sameh Askar. "Multigrid Methods for the Solution of Nonlinear Variational Inequalities." European Journal of Pure and Applied Mathematics 16, no. 3 (July 30, 2023): 1956–69. http://dx.doi.org/10.29020/nybg.ejpam.v16i3.4835.

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In this research, we investigate the numerical solution of second member problems that depend on the solution obtained through a multigrid method. Specifically, we focus on the application of multigrid techniques for solving nonlinear variational inequalities. The main objective is to establish the uniform convergence of the multigrid algorithm. To achieve this, we employ elementary subdifferential calculus and draw insights from the convergence theory of nonlinear multigrid methods.
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13

Мартыненко, С. И. "Numerical methods for black box software." Numerical Methods and Programming (Vychislitel'nye Metody i Programmirovanie), no. 2 (March 28, 2019): 147–69. http://dx.doi.org/10.26089/nummet.v20r215.

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Сформулированы требования к вычислительным алгоритмам для перспективного программного обеспечения, устроенного по принципу "черного ящика" и предназначенного для математического моделирования в механике сплошных сред. Выполнен анализ прикладных свойств классических многосеточных методов и универсальной многосеточной технологии в рамках проблемы "универсальность-эффективность-параллелизм". Показано, что близкая к оптимальной трудоемкость при минимуме проблемно-зависимых компонентов и высокая эффективность параллелизма достижимы при использовании универсальной многосеточной технологии на глобально структурированных сетках. Применение неструктурированных сеток потребует определения двух проблемно-зависимых компонентов (межсеточных операторов), которые значительно влияют на трудоемкость алгоритма. A number of requirements are formulated to the numerical algorithms for black box software intended for mathematical modeling in continuum mechanics. An analysis of applied properties of the classical multigrid methods and robust multigrid technique in the framework of "robustness-efficiency-parallelism" problem is performed. It is shown that a close-to-optimal complexity with the least number of problem-dependent components and high parallel efficiency can be achieved with the robust multigrid technique on globally structured grids. Application of unstructured grids requires the accurate definition of two problem-dependent components (intergrid operators) that strongly affect on the complexity of an algorithm.
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14

Mandel, Jan, Steve McCormick, and John Ruge. "An Algebraic Theory for Multigrid Methods for Variational Problems." SIAM Journal on Numerical Analysis 25, no. 1 (February 1988): 91–110. http://dx.doi.org/10.1137/0725008.

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15

Schöberl, Joachim, and Rolf Stenberg. "Multigrid Methods for a Stabilized Reissner–Mindlin Plate Formulation." SIAM Journal on Numerical Analysis 47, no. 4 (January 2009): 2735–51. http://dx.doi.org/10.1137/060672182.

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16

Xu, Xuefeng, and Chen-Song Zhang. "On the Ideal Interpolation Operator in Algebraic Multigrid Methods." SIAM Journal on Numerical Analysis 56, no. 3 (January 2018): 1693–710. http://dx.doi.org/10.1137/17m1162779.

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17

Bittencourt, Marco L., Craig C. Douglas, and Raúl A. Feijóo. "Nonnested multigrid methods for linear problems." Numerical Methods for Partial Differential Equations 17, no. 4 (July 2001): 313–31. http://dx.doi.org/10.1002/num.1013.

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18

Donatelli, Marco, Carlo Garoni, Carla Manni, Stefano Serra-Capizzano, and Hendrik Speleers. "Symbol-Based Multigrid Methods for Galerkin B-Spline Isogeometric Analysis." SIAM Journal on Numerical Analysis 55, no. 1 (January 2017): 31–62. http://dx.doi.org/10.1137/140988590.

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19

Brenner, Susanne C., Jintao Cui, and Li-yeng Sung. "Multigrid Methods Based on Hodge Decomposition for a Quad-Curl Problem." Computational Methods in Applied Mathematics 19, no. 2 (April 1, 2019): 215–32. http://dx.doi.org/10.1515/cmam-2019-0011.

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AbstractIn this paper we investigate multigrid methods for a quad-curl problem on graded meshes. The approach is based on the Hodge decomposition. The solution for the quad-curl problem is approximated by solving standard second-order elliptic problems and optimal error estimates are obtained on graded meshes. We prove the uniform convergence of the multigrid algorithm for the resulting discrete problem. The performance of these methods is illustrated by numerical results.
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20

Peisker, P., W. Rust, and E. Stein. "Iterative Solution Methods for Plate Bending Problems: Multigrid and Preconditionedcgalgorithm." SIAM Journal on Numerical Analysis 27, no. 6 (December 1990): 1450–65. http://dx.doi.org/10.1137/0727084.

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21

Hsiao, George C., and Shangyou Zhang. "Optimal Order Multigrid Methods for Solving Exterior Boundary Value Problems." SIAM Journal on Numerical Analysis 31, no. 3 (June 1994): 680–94. http://dx.doi.org/10.1137/0731036.

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22

Cai, Zhiqiang, Jan Mandel, and Steve McCormick. "Multigrid Methods for Nearly Singular Linear Equations and Eigenvalue Problems." SIAM Journal on Numerical Analysis 34, no. 1 (February 1997): 178–200. http://dx.doi.org/10.1137/s1064827594261139.

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23

Goldstein, Charles I. "Multigrid analysis of finite element methods with numerical integration." Mathematics of Computation 56, no. 194 (May 1, 1991): 409. http://dx.doi.org/10.1090/s0025-5718-1991-1066832-7.

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24

Yserentant, Harry. "Old and new convergence proofs for multigrid methods." Acta Numerica 2 (January 1993): 285–326. http://dx.doi.org/10.1017/s0962492900002385.

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Multigrid methods are the fastest known methods for the solution of the large systems of equations arising from the discretization of partial differential equations. For self-adjoint and coercive linear elliptic boundary value problems (with Laplace's equation and the equations of linear elasticity as two typical examples), the convergence theory reached a mature, if not its final state. The present article reviews old and new developments for this type of equation and describes the recent advances.
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25

McCormick, S. F. "Multigrid Methods for Variational Problems: General Theory for the V-Cycle." SIAM Journal on Numerical Analysis 22, no. 4 (August 1985): 634–43. http://dx.doi.org/10.1137/0722039.

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26

Kang, Kab S. "On the Finite Volume Multigrid Method: Comparison of Intergrid Transfer Operators." Computational Methods in Applied Mathematics 15, no. 2 (April 1, 2015): 189–202. http://dx.doi.org/10.1515/cmam-2014-0030.

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AbstractIn this paper, we consider finite volume multigrid methods on triangular meshes with control volume based intergrid transfer operators. We review the error analysis of the finite volume methods and the convergence analysis on the multigrid method. For several different triangulations, we investigate the error reduction factors of the multigrid method as a solver, and also as a preconditioner in the Preconditioned CGM and GMRES solvers. We also study the scaling properties of the finite volume multigrid method on a High Performance Computer. We identify that the intergrid transfer operator based on the trial function space has the best properties.
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27

Brannick, James, Xiaozhe Hu, Carmen Rodrigo, and Ludmil Zikatanov. "Local Fourier Analysis of Multigrid Methods with Polynomial Smoothers and Aggressive Coarsening." Numerical Mathematics: Theory, Methods and Applications 8, no. 1 (February 2015): 1–21. http://dx.doi.org/10.4208/nmtma.2015.w01si.

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We focus on the study of multigrid methods with aggressive coarsening and polynomial smoothers for the solution of the linear systems corresponding to finite difference/element discretizations of the Laplace equation. Using local Fourier analysis we determineautomaticallythe optimal values for the parameters involved in defining the polynomial smoothers and achieve fast convergence of cycles with aggressive coarsening. We also present numerical tests supporting the theoretical results and the heuristic ideas. The methods we introduce are highly parallelizable and efficient multigrid algorithms on structured and semi-structured grids in two and three spatial dimensions.
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28

Lee, Chang-Ock. "Multigrid Methods for the Pure Traction Problem of Linear Elasticity: Mixed Formulation." SIAM Journal on Numerical Analysis 35, no. 1 (February 1998): 121–45. http://dx.doi.org/10.1137/s0036142995282832.

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29

Reusken, Arnold. "On Maximum Norm Convergence of Multigrid Methods for Elliptic Boundary Value Problems." SIAM Journal on Numerical Analysis 31, no. 2 (April 1994): 378–92. http://dx.doi.org/10.1137/0731020.

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30

Beuchler, Sven. "Multigrid Solver for the Inner Problem in Domain Decomposition Methods for p-FEM." SIAM Journal on Numerical Analysis 40, no. 3 (January 2002): 928–44. http://dx.doi.org/10.1137/s0036142901393851.

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31

Reusken, Arnold. "On Maximum Norm Convergence of Multigrid Methods for Two-Point Boundary Value Problems." SIAM Journal on Numerical Analysis 29, no. 6 (December 1992): 1569–78. http://dx.doi.org/10.1137/0729090.

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32

Dahmen, Wolfgang. "Wavelet and multiscale methods for operator equations." Acta Numerica 6 (January 1997): 55–228. http://dx.doi.org/10.1017/s0962492900002713.

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More than anything else, the increase of computing power seems to stimulate the greed for tackling ever larger problems involving large-scale numerical simulation. As a consequence, the need for understanding something like the intrinsic complexity of a problem occupies a more and more pivotal position. Moreover, computability often only becomes feasible if an algorithm can be found that is asymptotically optimal. This means that storage and the number of floating point operations needed to resolve the problem with desired accuracy remain proportional to the problem size when the resolution of the discretization is refined. A significant reduction of complexity is indeed often possible, when the underlying problem admits a continuous model in terms of differential or integral equations. The physical phenomena behind such a model usually exhibit characteristic features over a wide range of scales. Accordingly, the most successful numerical schemes exploit in one way or another the interaction of different scales of discretization. A very prominent representative is the multigrid methodology; see, for instance, Hackbusch (1985) and Bramble (1993). In a way it has caused a breakthrough in numerical analysis since, in an important range of cases, it does indeed provide asymptotically optimal schemes. For closely related multilevel techniques and a unified treatment of several variants, such as multiplicative or additive subspace correction methods, see Bramble, Pasciak and Xu (1990), Oswald (1994), Xu (1992), and Yserentant (1993). Although there remain many unresolved problems, multigrid or multilevel schemes in the classical framework of finite difference and finite element discretizations exhibit by now a comparatively clear profile. They are particularly powerful for elliptic and parabolic problems.
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33

Borzì, Alfio, and Giuseppe Borzì. "Algebraic multigrid methods for solving generalized eigenvalue problems." International Journal for Numerical Methods in Engineering 65, no. 8 (February 19, 2006): 1186–96. http://dx.doi.org/10.1002/nme.1478.

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34

Evstigneev, Nikolay M. "Numerical analysis of Krylov multigrid methods for stationary advection-diffusion equation." Journal of Physics: Conference Series 1391 (November 2019): 012080. http://dx.doi.org/10.1088/1742-6596/1391/1/012080.

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35

TAGAMI, Daisuke, and Shin-ichiro SUGIMOTO. "Numerical Analysis of Multigrid Balancing Domain Decomposition Methods for Magnetostatic Problems." Proceedings of The Computational Mechanics Conference 2016.29 (2016): 4_301. http://dx.doi.org/10.1299/jsmecmd.2016.29.4_301.

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36

Martynenko, S. I. "Robust Multigrid Technique for Black Box Software." Computational Methods in Applied Mathematics 6, no. 4 (2006): 413–35. http://dx.doi.org/10.2478/cmam-2006-0026.

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Abstract This paper presents a new robust multigrid technique for solving boundary value problems in a black box manner. To overcome the problem of robustness, the technique is based on the incorporation of adaptation of boundary value problems to numerical methods, control volume discretization and a new multigrid solver into a united computational algorithm. The special multiple coarse grid correction strategy makes it possible to obtain problem-independent transfer operators. As a result, most modes are approximated on coarse grids to make the task of the smoother on the finest grid the least demanding. A detailed description of the robust multigrid technique and examples of its application for solving benchmark problems are given in the paper.
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37

Oosterlee, C. W., and F. J. Gaspar. "Multigrid relaxation methods for systems of saddle point type." Applied Numerical Mathematics 58, no. 12 (December 2008): 1933–50. http://dx.doi.org/10.1016/j.apnum.2007.11.014.

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38

Vandewalle, S., and G. Horton. "Fourier mode analysis of the multigrid waveform relaxation and time-parallel multigrid methods." Computing 54, no. 4 (December 1995): 317–30. http://dx.doi.org/10.1007/bf02238230.

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39

Köster, M., and S. Turek. "The Influence of Higher Order FEM Discretisations on Multigrid Convergence." Computational Methods in Applied Mathematics 6, no. 2 (2006): 221–32. http://dx.doi.org/10.2478/cmam-2006-0011.

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AbstractQuadratic and even higher order finite elements are interesting candidates for the numerical solution of partial differential equations (PDEs) due to their improved approximation properties in comparison to linear approaches. The systems of equations that arise from the discretisation of the underlying (elliptic) PDEs are often solved by iterative solvers like preconditioned Krylow-space methods, while multigrid solvers are still rarely used – which might be caused by the high effort that is associated with the realisation of the necessary data structures as well as smoothing and intergrid transfer operators. In this note, we discuss the numerical analysis of quadratic conforming finite elements in a multigrid solver. Using the “correct” grid transfer operators in conjunction with a quadratic finite element approximation allows to formulate an improved approximation property which enhances the (asymptotic) behaviour of multigrid: If m denotes the number of smoothing steps, the convergence rates behave asymptotically like O(1/m2) in contrast to O(1/m) for linear FEM.
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40

Liao, Jia, Ting-Zhu Huang, and Bruno Carpentieri. "Two novel aggregation-based algebraic multigrid methods." Miskolc Mathematical Notes 14, no. 1 (2013): 143. http://dx.doi.org/10.18514/mmn.2013.344.

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41

Takacs, Stefan, and Walter Zulehner. "Convergence Analysis of All-at-Once Multigrid Methods for Elliptic Control Problems under Partial Elliptic Regularity." SIAM Journal on Numerical Analysis 51, no. 3 (January 2013): 1853–74. http://dx.doi.org/10.1137/120880884.

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42

Bolten, Matthias, Thomas K. Huckle, and Christos D. Kravvaritis. "Sparse matrix approximations for multigrid methods." Linear Algebra and its Applications 502 (August 2016): 58–76. http://dx.doi.org/10.1016/j.laa.2015.11.008.

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43

Fischer, Rainer, and Thomas Huckle. "Multigrid methods for anisotropic BTTB systems." Linear Algebra and its Applications 417, no. 2-3 (September 2006): 314–34. http://dx.doi.org/10.1016/j.laa.2006.02.032.

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44

Martynenko, S. I. "Potentialities of the Robust Multigrid Technique." Computational Methods in Applied Mathematics 10, no. 1 (2010): 87–94. http://dx.doi.org/10.2478/cmam-2010-0004.

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AbstractThe present paper discusses the parallelization of the robust multigrid technique (RMT) and the possible way of applying this to unstructured grids. As opposed to the classical multigrid methods, the RMT is a trivial method of parallelization on coarse grids independent of the smoothing iterations. Estimates of the minimum speed-up and parallelism efficiency are given. An almost perfect load balance is demonstrated in a 3D illustrative test. To overcome the geometric nature of the technique, the RMT is used as a preconditioner in solving PDEs on unstructured grids. The procedure of auxiliary structured grids generation is considered in details.
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45

Lu, Peipei, Andreas Rupp, and Guido Kanschat. "Analysis of Injection Operators in Geometric Multigrid Solvers for HDG Methods." SIAM Journal on Numerical Analysis 60, no. 4 (August 2022): 2293–317. http://dx.doi.org/10.1137/21m1400110.

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46

Gräser, Carsten, Max Kahnt, and Ralf Kornhuber. "Numerical Approximation of Multi-Phase Penrose–Fife Systems." Computational Methods in Applied Mathematics 16, no. 4 (October 1, 2016): 523–42. http://dx.doi.org/10.1515/cmam-2016-0020.

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AbstractWe consider a non-isothermal multi-phase field model. We subsequently discretize implicitly in time and with linear finite elements. The arising algebraic problem is formulated in two variables where one is the multi-phase field, and the other contains the inverse temperature field. We solve this saddle point problem numerically by a non-smooth Schur–Newton approach using truncated non-smooth Newton multigrid methods. An application in grain growth as occurring in liquid phase crystallization of silicon is considered.
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47

Langer, U., and D. Pusch. "Data-sparse algebraic multigrid methods for large scale boundary element equations." Applied Numerical Mathematics 54, no. 3-4 (August 2005): 406–24. http://dx.doi.org/10.1016/j.apnum.2004.09.011.

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48

Gaspar, F. J., J. L. Gracia, F. J. Lisbona, and C. Rodrigo. "On geometric multigrid methods for triangular grids using three-coarsening strategy." Applied Numerical Mathematics 59, no. 7 (July 2009): 1693–708. http://dx.doi.org/10.1016/j.apnum.2009.01.003.

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49

Tu, J. Y., and L. Fuchs. "Calculation of flows using three-dimensional overlapping grids and multigrid methods." International Journal for Numerical Methods in Engineering 38, no. 2 (January 30, 1995): 259–82. http://dx.doi.org/10.1002/nme.1620380207.

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50

Bolten, Matthias, Stephanie Friedhoff, Andreas Frommer, Matthias Heming, and Karsten Kahl. "Algebraic multigrid methods for Laplacians of graphs." Linear Algebra and its Applications 434, no. 11 (June 2011): 2225–43. http://dx.doi.org/10.1016/j.laa.2010.11.008.

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