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Multigrid Methods
Multigrid Methods Volker John Winter Semester 2013/14 Contents 1 Literature 2 2 Model Problems 3 3 Detailed Investigation of Classical Iterative Schemes 8 3.1 General Aspects of Classical Iterative Schemes . .8 3.2 The Jacobi and Damped Jacobi Method . 10 3.3 The Gauss{Seidel Method and the SOR Method . 15 3.4 Summary . 18 4 Grid Transfer 19 4.1 Algorithms with Coarse Grid Systems, the Residual Equation . 19 4.2 Prolongation or Interpolation . 21 4.3 Restriction . 23 5 The Two-Level Method 27 5.1 The Coarse Grid Problem . 27 5.2 General Approach for Proving the Convergence of the Two-Level Method . 30 5.3 The Smoothing Property of the Damped Jacobi Iteration . 32 5.4 The Approximation Property . 33 5.5 Summary . 36 6 The Multigrid Method 37 6.1 Multigrid Cycles . 37 6.2 Convergence of the W-cycle . 39 6.3 Computational Work of the Multigrid γ-Cycle . 45 7 Algebraic Multigrid Methods 49 7.1 Components of an AMG and Definitions . 49 7.2 Algebraic Smoothness . 52 7.3 Coarsening . 56 7.4 Prolongation . 58 7.5 Concluding Remarks . 60 8 Outlook 61 1 Chapter 1 Literature Remark 1.1 Literature. There are several text books about multigrid methods, e.g., • Briggs et al. (2000), easy to read introduction, • Hackbusch (1985), the classical book, sometimes rather hard to read, • Shaidurov (1995), • Wesseling (1992), an introductionary book, • Trottenberg et al. (2001). 2 2 Chapter 2 Model Problems Remark 2.1 Motivation. The basic ideas and properties of multigrid methods will be explained in this course on two model problems. -
Hybrid Particle-Grid Water Simulation Using Multigrid Pressure Solver Per Karlsson
LiU-ITN-TEK-G--14/006-SE Hybrid Particle-Grid Water Simulation using Multigrid Pressure Solver Per Karlsson 2014-03-13 Department of Science and Technology Institutionen för teknik och naturvetenskap Linköping University Linköpings universitet nedewS ,gnipökrroN 47 106-ES 47 ,gnipökrroN nedewS 106 47 gnipökrroN LiU-ITN-TEK-G--14/006-SE Hybrid Particle-Grid Water Simulation using Multigrid Pressure Solver Examensarbete utfört i Medieteknik vid Tekniska högskolan vid Linköpings universitet Per Karlsson Handledare George Baravdish Examinator Camilla Forsell Norrköping 2014-03-13 Upphovsrätt Detta dokument hålls tillgängligt på Internet – eller dess framtida ersättare – under en längre tid från publiceringsdatum under förutsättning att inga extra- ordinära omständigheter uppstår. Tillgång till dokumentet innebär tillstånd för var och en att läsa, ladda ner, skriva ut enstaka kopior för enskilt bruk och att använda det oförändrat för ickekommersiell forskning och för undervisning. Överföring av upphovsrätten vid en senare tidpunkt kan inte upphäva detta tillstånd. All annan användning av dokumentet kräver upphovsmannens medgivande. För att garantera äktheten, säkerheten och tillgängligheten finns det lösningar av teknisk och administrativ art. Upphovsmannens ideella rätt innefattar rätt att bli nämnd som upphovsman i den omfattning som god sed kräver vid användning av dokumentet på ovan beskrivna sätt samt skydd mot att dokumentet ändras eller presenteras i sådan form eller i sådant sammanhang som är kränkande för upphovsmannens litterära eller konstnärliga anseende eller egenart. För ytterligare information om Linköping University Electronic Press se förlagets hemsida http://www.ep.liu.se/ Copyright The publishers will keep this document online on the Internet - or its possible replacement - for a considerable time from the date of publication barring exceptional circumstances. -
Multigrid Algorithms for High Order Discontinuous Galerkin Methods
Multigrid algorithms for high order discontinuous Galerkin methods Paola F. Antonietti1, Marco Sarti2, and Marco Verani3 Abstract I n this paper we study the performance of h- and p-multigrid algorithms for high order Discontinuous Galerkin discretizations of elliptic problems. We test the performance of the multigrid schemes employing a wide class of smoothers and considering both two- and three-dimensional test cases. 1 Introduction In the framework of multigrid solvers for Discontinuous Galerkin (DG) schemes, the first contributions are due to Gopalakrishnan and Kanschat [16] and Brenner and Zhao [10]. In [16] a V-cycle preconditioner for a Sym- metric Interior Penalty (SIP) discretization of an elliptic problem is analyzed. They prove that the condition number of the preconditioned system is uni- formly bounded with respect to the mesh size and the number of levels. In [10] V-cycle, F-cycle and W-cycle multigrid schemes for SIP discretizations are presented and analyzed, employing the additive theory developed in [8, 9]. A uniform bound for the error propagation operator is shown provided the number of smoothing steps is large enough. All the previously cited works focus on low order, i.e., linear, DG approximations. With regard to high order DG discretizations, h- and p-multigrid schemes are successfully em- ployed for the numerical solution of many different kinds of problems, see e.g. [14, 20, 22, 21, 24, 6], even if only few theoretical results are available that show the convergence properties of the underlying algorithms. In the MOX, Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano ITALY. -
Hybrid Multigrid/Schwarz Algorithms for the Spectral Element Method
Hybrid Multigrid/Schwarz Algorithms for the Spectral Element Method James W. Lottes¤ and Paul F. Fischery February 4, 2004 Abstract We study the performance of the multigrid method applied to spectral element (SE) discretizations of the Poisson and Helmholtz equations. Smoothers based on finite element (FE) discretizations, overlapping Schwarz methods, and point-Jacobi are con- sidered in conjunction with conjugate gradient and GMRES acceleration techniques. It is found that Schwarz methods based on restrictions of the originating SE matrices converge faster than FE-based methods and that weighting the Schwarz matrices by the inverse of the diagonal counting matrix is essential to effective Schwarz smoothing. Sev- eral of the methods considered achieve convergence rates comparable to those attained by classic multigrid on regular grids. 1 Introduction The availability of fast elliptic solvers is essential to many areas of scientific computing. For unstructured discretizations in three dimensions, iterative solvers are generally optimal from both work and storage standpoints. Ideally, one would like to have computational complexity that scales as O(n) for an n-point grid problem in lRd, implying that the it- eration count should be bounded as the mesh is refined. Modern iterative methods such as multigrid and Schwarz-based domain decomposition achieve bounded iteration counts through the introduction of multiple representations of the solution (or the residual) that allow efficient elimination of the error at each scale. The theory for these methods is well established for classical finite difference (FD) and finite element (FE) discretizations, and order-independent convergence rates are often attained in practice. For spectral element (SE) methods, there has been significant work on the development of Schwarz-based methods that employ a combination of local subdomain solves and sparse global solves to precondition conjugate gradient iteration. -
Multigrid and Saddle-Point Preconditioners for Unfitted Finite
Multigrid and saddle-point preconditioners for unfitted finite element modelling of inclusions Hardik Kothari∗and Rolf Krause† Institute of Computational Science, Universit`adella Svizzera italiana, Lugano, Switzerland February 17, 2021 Abstract In this work, we consider the modeling of inclusions in the material using an unfitted finite element method. In the unfitted methods, structured background meshes are used and only the underlying finite element space is modified to incorporate the discontinuities, such as inclusions. Hence, the unfitted methods provide a more flexible framework for modeling the materials with multiple inclusions. We employ the method of Lagrange multipliers for enforcing the interface conditions between the inclusions and matrix, this gives rise to the linear system of equations of saddle point type. We utilize the Uzawa method for solving the saddle point system and propose preconditioning strategies for primal and dual systems. For the dual systems, we review and compare the preconditioning strategies that are developed for FETI and SIMPLE methods. While for the primal system, we employ a tailored multigrid method specifically developed for the unfitted meshes. Lastly, the comparison between the proposed preconditioners is made through several numerical experiments. Keywords: Unfitted finite element method, multigrid method, saddle-point problem 1 Introduction In the modeling of many real-world engineering problems, we encounter material discontinuities, such as inclusions. The inclusions are found naturally in the materials, or can be artificially introduced to produce desired mechanical behavior. The finite element (FE) modeling of such inclusions requires to generate meshes that can resolve the interface between the matrix and inclusions, which can be a computationally cumbersome and expensive task. -
Multigrid Method Based on a Space-Time Approach with Standard Coarsening for Parabolic Problems
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CWI's Institutional Repository Applied Mathematics and Computation 317 (2018) 25–34 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc Multigrid method based on a space-time approach with standard coarsening for parabolic problems Sebastião Romero Franco a,b, Francisco José Gaspar c, Marcio Augusto Villela ∗ Pinto d, Carmen Rodrigo e, a Department of Mathematics, State University of Centro-Oeste, Irati, PR, Brazil b Graduate Program in Numerical Methods in Engineering, Federal University of Paraná, Curitiba, PR, Brazil c Centrum Wiskunde & Informatica (CWI), Science Park 123, P.O. Box 94079, Amsterdam 1090, The Netherlands d Department of Mechanical Engineering, Federal University of Paraná, Curitiba, PR, Brazil e IUMA and Applied Mathematics Department, University of Zaragoza, María de Luna, 3, Zaragoza 50018, Spain a r t i c l e i n f o a b s t r a c t MSC: In this work, a space-time multigrid method which uses standard coarsening in both tem- 00-01 poral and spatial domains and combines the use of different smoothers is proposed for the 99-00 solution of the heat equation in one and two space dimensions. In particular, an adaptive Keywords: smoothing strategy, based on the degree of anisotropy of the discrete operator on each Space-time multigrid grid-level, is the basis of the proposed multigrid algorithm. Local Fourier analysis is used Local Fourier analysis for the selection of the crucial parameter defining such an adaptive smoothing approach. -
Hp-Multigrid As Smoother Algorithm for Higher Order Discontinuous Galerkin Discretizations of Advection Dominated Flows Part I
HP-MULTIGRID AS SMOOTHER ALGORITHM FOR HIGHER ORDER DISCONTINUOUS GALERKIN DISCRETIZATIONS OF ADVECTION DOMINATED FLOWS PART I. MULTILEVEL ANALYSIS J.J.W. VAN DER VEGT∗ AND S. RHEBERGENy AMS subject classifications. Primary 65M55; Secondary 65M60, 76M10. Abstract. The hp-Multigrid as Smoother algorithm (hp-MGS) for the solution of higher order accurate space-(time) discontinuous Galerkin discretizations of advection dominated flows is pre- sented. This algorithm combines p-multigrid with h-multigrid at all p-levels, where the h-multigrid acts as smoother in the p-multigrid. The performance of the hp-MGS algorithm is further improved using semi-coarsening in combination with a new semi-implicit Runge-Kutta method as smoother. A detailed multilevel analysis of the hp-MGS algorithm is presented to obtain more insight into the theoretical performance of the algorithm. As model problem a fourth order accurate space-time dis- continuous Galerkin discretization of the advection-diffusion equation is considered. The multilevel analysis shows that the hp-MGS algorithm has excellent convergence rates, both for low and high cell Reynolds numbers and on highly stretched meshes. Key words. multigrid algorithms, discontinuous Galerkin methods, higher order accurate dis- cretizations, space-time methods, Runge-Kutta methods, Fourier analysis, multilevel analysis 1. Introduction. Discontinuous Galerkin finite element methods are well suited to obtain higher order accurate discretizations on unstructured meshes. The use of basis functions which are only weakly coupled to neighboring elements results in a local discretization which allows, in combination with hp-mesh adaptation, the efficient capturing of detailed structures in the solution, and is also beneficial for parallel computing. -
CONVERGENCE of ALGEBRAIC MULTIGRID BASED on SMOOTHED AGGREGATION∗ ˇ PETR VANEK† , MARIAN BREZINA‡ , and JAN MANDEL§ February 1998 Revised February 2000 Abstract
CONVERGENCE OF ALGEBRAIC MULTIGRID BASED ON SMOOTHED AGGREGATION∗ ˇ PETR VANEKy , MARIAN BREZINAz , AND JAN MANDELx February 1998 Revised February 2000 Abstract. We prove an abstract convergence estimate for the Algebraic Multigrid Method with prolongator defined by a disaggregation followed by a smoothing. The method input is the problem matrix and a matrix of the zero energy modes of the same problem but with natural boundary conditions. The construction is described in the case of a general elliptic system. The condition number bound increases only as a polynomial of the number of levels, and requires only a uniform weak approximation property for the aggregation operators. This property can be a-priori verified computationally once the aggregates are known. For illustration, it is also verified here for a uniformly elliptic diffusion equations discretized by linear conforming quasiuniform finite elements. Only very weak and natural assumptions on the hierarchy of aggregates are needed. Key words. Algebraic multigrid, zero energy modes, convergence theory, computational mechanics, finite elements, iterative solvers. 1. Introduction. This paper establishes abstract convergence bounds for an Algebraic Multigrid Method (AMG) based on smoothed aggregation. The bounds are obtained by invoking the general convergence theory of [6]. Our main result is a bound on the condition number that grows only as a power of the number of levels, and requires only a weak approximation property for the aggregates, similar to the weak approximation condition in classical AMG investigations [9, 22, 28, 37]. Our weak approximation condition can be easily verified computationally, and we show that it holds for general unstructured meshes and under natural assumptions on aggregates used to construct the coarse levels. -
Multigrid Solvers for Immersed Finite Element Methods and Immersed Isogeometric Analysis
Computational Mechanics https://doi.org/10.1007/s00466-019-01796-y ORIGINAL PAPER Multigrid solvers for immersed finite element methods and immersed isogeometric analysis F. de Prenter1,3 · C. V. Verhoosel1 · E. H. van Brummelen1 · J. A. Evans2 · C. Messe2,4 · J. Benzaken2,5 · K. Maute2 Received: 26 March 2019 / Accepted: 10 November 2019 © The Author(s) 2019 Abstract Ill-conditioning of the system matrix is a well-known complication in immersed finite element methods and trimmed isogeo- metric analysis. Elements with small intersections with the physical domain yield problematic eigenvalues in the system matrix, which generally degrades efficiency and robustness of iterative solvers. In this contribution we investigate the spectral prop- erties of immersed finite element systems treated by Schwarz-type methods, to establish the suitability of these as smoothers in a multigrid method. Based on this investigation we develop a geometric multigrid preconditioner for immersed finite element methods, which provides mesh-independent and cut-element-independent convergence rates. This preconditioning technique is applicable to higher-order discretizations, and enables solving large-scale immersed systems at a computational cost that scales linearly with the number of degrees of freedom. The performance of the preconditioner is demonstrated for conventional Lagrange basis functions and for isogeometric discretizations with both uniform B-splines and locally refined approximations based on truncated hierarchical B-splines. Keywords Immersed finite element method · Fictitious domain method · Iterative solver · Preconditioner · Multigrid 1 Introduction [14–21], scan based analysis [22–27] and topology optimiza- tion, e.g., [28–34]. Immersed methods are useful tools to avoid laborious and An essential aspect of finite element methods and iso- computationally expensive procedures for the generation of geometric analysis is the computation of the solution to a body-fitted finite element discretizations or analysis-suitable system of equations. -
HMG--Homogeneous Multigrid For
HMG — HOMOGENEOUS MULTIGRID FOR HDG PEIPEI LU, ANDREAS RUPP, AND GUIDO KANSCHAT Abstract. We introduce a homogeneous multigrid method in the sense that it uses the same HDG discretization scheme for Poisson’s equation on all levels. In particular, we construct a stable injection operator and prove optimal convergence of the method under the assumption of elliptic regularity. Numerical experiments underline our analytical findings. 1. Introduction While hybridizable discontinuous Galerkin (HDG) methods have be- come popular in mathematics and applications over the last years, lit- erature on efficient solution of the resulting discrete systems is still scarce. In this article, we propose to our knowledge the first multigrid preconditioner for such methods which is homogeneous in the sense that it uses the same discretization scheme on all levels. Such methods are important, since they have the same mathematical properties on all levels. They can also be advantageous from a computational point of view, since their data structures and execution patterns are more regular. HDG methods have been gaining popularity in the last decade. Orig- inally, they were analyzed for the Laplacian, see for instance [12] for an overview. Meanwhile, they have been applied to stationary [11, 14, 15, 17, 28, 29] and instationary [25] Stokes systems, to the locking-free discretization of problems in elasticity [19], as well as to plates [9, 24] and beams [7]. However, only few results have been achieved for solv- ing the large systems of equations arising from this type of numerical method, while multigrid [21] and domain decomposition [18] solvers arXiv:2011.14018v1 [math.NA] 27 Nov 2020 for earlier discontinuous Galerkin (DG) schemes have been available for many years. -
An Efficient Finite Element Based Multigrid Method for Simulations Of
An efficient finite element based multigrid method for simulations of the mechanical behavior of heterogeneous materials using CT images Xiaodong Liu, Julien Réthoré, Marie-Christine Baietto, Philippe Sainsot, Antonius Lubrecht To cite this version: Xiaodong Liu, Julien Réthoré, Marie-Christine Baietto, Philippe Sainsot, Antonius Lubrecht. An effi- cient finite element based multigrid method for simulations of the mechanical behavior of heterogeneous materials using CT images. Computational Mechanics, Springer Verlag, 2020, 10.1007/s00466-020- 01909-y. hal-02919095 HAL Id: hal-02919095 https://hal.archives-ouvertes.fr/hal-02919095 Submitted on 21 Aug 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Distributed under a Creative Commons Attribution| 4.0 International License An efficient finite element based multigrid method for simulations of the mechanical behavior of heterogeneous materials using CT images Xiaodong Liu1, Julien R´ethor´e∗1, Marie-Christine Baietto2, Philippe Sainsot2, and Antonius Adrianus Lubrecht2 1GeM, CNRS UMR 6183, Centrale Nantes, F-44321, France 2Univ Lyon, INSA-Lyon, CNRS UMR5259, LaMCoS, F-69621, France 19 August, 2020 Abstract X-ray tomography techniques give researchers the full access to material inner structures. With such ample information, employing numerical simulation on real material images becomes more and more common. -
Fast Eigenpairs Computation with Operator Adapted Wavelets and Hierarchical Subspace Correction
Fast eigenpairs computation with operator adapted wavelets and hierarchical subspace correction Hehu Xie,∗ Lei Zhang,y Houman Owhadiz September 5, 2019 We present a method for the fast computation of the eigenpairs of a bijective positive symmetric linear operator L. The method is based on a combination of operator adapted wavelets (gamblets) with hierarchical subspace correction. First, gamblets provide a raw but fast approximation of the eigensubspaces of L by block-diagonalizing L into sparse and well-conditioned blocks. Next, the hierarchical subspace correction method, computes the eigenpairs associ- ated with the Galerkin restriction of L to a coarse (low dimensional) gamblet subspace, and then, corrects those eigenpairs by solving a hierarchy of linear problems in the finer gamblet subspaces (from coarse to fine, using multigrid iteration). The proposed algorithm is robust to the presence of multiple (a con- tinuum of) scales and is shown to be of near-linear complexity when L is an s −s (arbitrary local, e.g. differential) operator mapping H0(Ω) to H (Ω) (e.g. an elliptic PDE with rough coefficients). Keywords. Multiscale eigenvalue problem, gamblet decomposition, multi- grid iteration, subspace correction, numerical homogenization. AMS subject classifications. 65N30, 65N25, 65L15, 65B99. 1 Introduction arXiv:1806.00565v4 [math.NA] 4 Sep 2019 Solving large scale eigenvalue problems is one of the most fundamental and challenging tasks in modern science and engineering. Although high-dimensional eigenvalue problems ∗[email protected]. LSEC, NCMIS, Institute of Computational Mathematics, Academy of Mathe- matics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China, and University of Chinese Academy of Sciences, Beijing 100049, China.