The Fourier analysis of the \emph{p}-multigrid acceleration technique is considered for a dual-time scheme applied to the advection-diffusion equation with various cycle configurations. It is found that improved convergence can be achieved through \emph{V}-cycle asymmetry where additional prolongation smoothing is applied. Experiments conducted on the artificial compressibility formulation of the Navier--Stokes equations found that these analytic findings could be observed numerically in the pressure residual, whereas velocity terms---which are more hyperbolic in character---benefited primarily from increased pseudo-time steps.
Multigrid methods are popular for solving linear systems derived from discretizing PDEs. Local Fourier Analysis (LFA) is a technique for investigating and tuning multigrid methods. P-multigrid is popular for high-order or spectral finite element methods, especially on unstructured meshes. In this paper, we introduce LFAToolkit.jl, a new Julia package for LFA of high-order finite element methods. LFAToolkit.jl analyzes preconditioning techniques for arbitrary systems of second order PDEs and supports mixed finite element methods. Specifically, we develop LFA of p-multigrid with arbitrary second…
In this paper we analyse the convergence properties of two-level, W-cycle and V-cycle agglomeration-based geometric multigrid schemes for the numerical solution of the linear system of equations stemming from the lowest order $C^0$-conforming Virtual Element discretization of two-dimensional second-order elliptic partial differential equations. The sequence of agglomerated tessellations are nested, but the corresponding multilevel virtual discrete spaces are generally non-nested thus resulting into non-nested multigrid algorithms. We prove the uniform convergence of the two-level method with r…
In this paper we present a Local Fourier Analysis of a space-time multigrid solver for a hyperbolic test problem. The space-time discretization is based on arbitrarily high order discontinuous Galerkin spectral element methods in time and a first order finite volume method in space. We apply a block Jacobi smoother and consider coarsening in space-time, as well as temporal coarsening only. Asymptotic convergence factors for the smoother and the two-grid method for both coarsening strategies are presented. For high CFL numbers, the convergence factors for both strategies are $0.5$ for first ord…
This note describes the full approximation storage (FAS) multigrid scheme for an easy one-dimensional nonlinear boundary value problem. The problem is discretized by a simple finite element (FE) scheme. We apply both FAS V-cycles and F-cycles, with a nonlinear Gauss-Seidel smoother, to solve the resulting finite-dimensional problem. The mathematics of the FAS restriction and prolongation operators, in the FE case, are explained. A self-contained Python program implements the scheme. Optimal performance, i.e. work proportional to the number of unknowns, is demonstrated for both kinds of cycles,…
Multigrid Methods for Hellan-Herrmann-Johnson Mixed Method of Kirchhoff Plate Bending Problems
University of California at Irvine（加州大学欧文分校）；Beijing Institute for Scientific and Engineering Computing, Beijing University of Technology（北京工业大学北京科学计算与工程研究所）；Peking University（北京大学）；College of Mathematics and Information Science, Wenzhou University（温州大学数学与信息科学学院）
We present an efficient, robust and fully GPU-accelerated aggregation-based algebraic multigrid preconditioning technique for the solution of large sparse linear systems. These linear systems arise from the discretization of elliptic PDEs. The method involves two stages, setup and solve. In the setup stage, hierarchical coarse grids are constructed through aggregation of the fine grid nodes. These aggregations are obtained using a set of maximal independent nodes from the fine grid nodes. We use a ``fine-grain'' parallel algorithm for finding a maximal independent set from a graph of strong ne…
Local Fourier analysis is a useful tool for predicting and analyzing the performance of many efficient algorithms for the solution of discretized PDEs, such as multigrid and domain decomposition methods. The crucial aspect of local Fourier analysis is that it can be used to minimize an estimate of the spectral radius of a stationary iteration, or the condition number of a preconditioned system, in terms of a symbol representation of the algorithm. In practice, this is a "minimax" problem, minimizing with respect to solver parameters the appropriate measure of work, which involves maximizing ov…
We propose and analyze a Vanka-type multigrid solver for solving a sequence of complex-shifted Laplacian systems arising in diagonalization-based parallel-in-time algorithms for evolutionary equations. Under suitable assumption, local Fourier analysis shows the proposed Vanka-type smoother achieves a uniform smoothing factor, which is verified by several numerical examples.
The solution to the Poisson equation arising from the spectral element discretization of the incompressible Navier-Stokes equation requires robust preconditioning strategies. One such strategy is multigrid. To realize the potential of multigrid methods, effective smoothing strategies are needed. Chebyshev polynomial smoothing proves to be an effective smoother. However, there are several improvements to be made, especially at the cost of symmetry. For the same cost per iteration, a full V-cycle with $k$ order Chebyshev polynomial smoothing may be substituted with a half V-cycle with order $2k$…
In this paper, we will construct and analyze a multigrid algorithm that can be applied to weighted H(div)-problems on a two-dimensional domain. These problems arise after performing a dimension reduction to a three-dimensional axisymmetric H(div)-problem. We will use recently developed Fourier finite element spaces that can be applied to axisymmetric H(div)-problems with general data. We prove that if the axisymmetric domain is convex, then the multigrid V-cycle with modern smoothers will converge uniformly with respect to the meshsize.
Multigrid methods are popular solution algorithms for many discretized PDEs, either as standalone iterative solvers or as preconditioners, due to their high efficiency. However, the choice and optimization of multigrid components such as relaxation schemes and grid-transfer operators is crucial to the design of optimally efficient algorithms. It is well--known that local Fourier analysis (LFA) is a useful tool to predict and analyze the performance of these components. In this paper, we develop a local Fourier analysis of monolithic multigrid methods based on additive Vanka relaxation schemes …
Automated Grammar-based Algebraic Multigrid Design With Evolutionary Algorithms
Friedrich-Alexander-Universität Erlangen-Nürnberg（埃朗根-纽伦堡弗里德里希-亚历山大大学）；Lawrence Livermore National Laboratory（劳伦斯利弗莫尔国家实验室）；CERFACS；Department of Applied Mathematics, VSB-Technical University of Ostrava（应用数学系，奥斯特拉瓦技术大学）
We propose V--cycle multigrid methods for vector field problems arising from the lowest order hexahedral Nédélec finite element. Since the conventional scalar smoothing techniques do not work well for the problems, a new type of smoothing method is necessary. We introduce new smoothers based on substructuring with nonoverlapping domain decomposition methods. We provide the convergence analysis and numerical experiments that support our theory.
This paper builds on the algebraic theory in the companion paper [Algebraic Error Analysis for Mixed-Precision Multigrid Solvers] to obtain discretization-error-accurate solutions for linear elliptic partial differential equations (PDEs) by mixed-precision multigrid solvers. It is often assumed that the achievable accuracy is limited by discretization or algebraic errors. On the contrary, we show that the quantization error incurred by simply storing the matrix in any fixed precision quickly begins to dominate the total error as the discretization is refined. We extend the existing theory to a…
Local Fourier analysis is a commonly used tool to assess the quality and aid in the construction of geometric multigrid methods for translationally invariant operators. In this paper we automate the process of local Fourier analysis and present a framework that can be applied to arbitrary, including non-orthogonal, repetitive structures. To this end we introduce the notion of crystal structures and a suitable definition of corresponding wave functions, which allow for a natural representation of almost all translationally invariant operators that are encountered in applications, e.g., discreti…
Scalable Multigrid Solver for the Helmholtz Equation: Real-Shifted Coarse Grid Correction
Institute for Interdisciplinary Computational Science, the Stein Faculty of Computer and Information Science, Ben-Gurion University of the Negev（内盖夫本-古里安大学斯坦计算机与信息科学学院跨学科计算科学研究所）
We present a Boundary Local Fourier Analysis (BLFA) to optimize the relaxation parameters of boundary conditions in a multigrid framework. The method is implemented to solve elliptic equations on curved domains embedded in a uniform Cartesian mesh, although it is designed to be extended for general PDEs in curved domains, wherever a multigrid technique can be implemented. The boundary is implicitly defined by a level-set function and a ghost-point technique is employed to treat the boundary conditions. Existing strategies in literature adopt a constant relaxation parameter on the whole boundar…
A fast multigrid solver is presented for high-order accurate Stokes problems discretised by local discontinuous Galerkin (LDG) methods. The multigrid algorithm consists of a simple V-cycle, using an element-wise block Gauss-Seidel smoother. The efficacy of this approach depends on the LDG pressure penalty stabilisation parameter -- provided the parameter is suitably chosen, numerical experiment shows that: (i) for steady-state Stokes problems, the convergence rate of the multigrid solver can match that of classical geometric multigrid methods for Poisson problems; (ii) for unsteady Stokes prob…
p-multigrid method for the discontinuous Galerkin discretization of elliptic problems
Hua Loo-Keng Center for Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院华罗庚数学科学研究院）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院计算数学与科学工程计算研究所LSEC）；School of Mathematical Science, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）
We expand the applicabilities and capabilities of an already existing space-time parallel method based on a block Jacobi smoother. First we formulate a more detailed criterion for spatial coarsening, which enables the method to deal with unstructured meshes and varying material parameters. Further we investigate the application to the eddy-current equation, where the non-trivial kernel of the curl operator causes severe problems. This is remedied with a new nodal auxiliary space correction. We proceed to identify convergence rates by local Fourier analysis and numerical experiments. Finally, w…
Mixed precision multigrid with smoothing based on incomplete Cholesky factorization
CNRS-IRIT（法国国家科学研究中心-信息学与系统技术研究所）；IFP Energies Nouvelles（法国石油研究院）；Department of Numerical Mathematics, Charles University（布拉格查理大学数值数学系）；Technical University of Munich - Campus Heilbronn（慕尼黑工业大学海尔布隆校区）；Ludwig-Maximilians-Universität München - Department of Earth and Environmental Sciences（慕尼黑路德维希-马克西米利安大学地球与环境科学系）；Department of Computer Science, FAU Erlangen-Nürnberg（埃尔朗根-纽伦堡大学计算机系）；Department of Applied Mathematics, VSB-Technical University of Ostrava（俄斯特拉发理工大学应用数学系）；Centre Européen de Recherche et de Formation Avancée en Calcul Scientifique（欧洲高级科学计算研究与培训中心）
