In this paper we study convergence estimates for a multigrid algorithm with smoothers of successive subspace correction (SSC) type, applied to symmetric elliptic PDEs. First, we revisit a general convergence analysis on a class of multigrid algorithms in a fairly general setting, where no regularity assumptions are made on the solution. In this framework, we are able to explicitly highlight the dependence of the multigrid error bound on the number of smoothing steps. For the case of no regularity assumptions, this represents a new addition to the existing theory. Then, we analyze successive subspace correction smoothing schemes for a set of uniform and local refinement applications with either nested or non-nested overlapping subdomains. For these applications, we explicitly derive bounds for the multigrid error, and identify sufficient conditions for these bounds to be independent of the number of multigrid levels. For the local refinement applications, finite element grids with arbitrary hanging nodes configurations are considered. The analysis of these smoothing schemes is cast within the far-reaching multiplicative Schwarz framework.
The solution of saddle-point problems, such as the Stokes equations, is a challenging task, especially in large-scale problems. Multigrid methods are one of the most efficient solvers for such systems of equations and can achieve convergence rates independent of the problem size. The smoother is a crucial component of multigrid methods and significantly affects its overall efficiency. We propose a Vanka-type smoother that we refer to as Restricted Additive Vanka and investigate its convergence in the context of adaptive geometric multigrid methods for the Stokes equations. The proposed smoothe…
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.
The dual formulation for linear elasticity, in contrast to the primal formulation, is not affected by locking, as it is based on the stresses as main unknowns. Thus it is quite attractive for nearly incompressible and incompressible materials. Discretization with mixed finite elements will lead to -- possibly large -- linear saddle point systems with a particular structure. Whereas efficient multigrid methods exist for solving problems in mixed plane elasticity, to the knowledge of the authors, no multigrid methods are readily available for the general dual formulation. Two are the main challe…
Overlapping block smoothers efficiently damp the error contributions from highly oscillatory components within multigrid methods for the Stokes equations but they are computationally expensive. This paper is concentrated on the development and analysis of new block smoothers for the Stokes equations that are discretized on staggered grids. These smoothers are non-overlapping and therefore desirable due to reduced computational costs. Traditional geometric multigrid methods are based on simple pointwise smoothers. However, the efficiency of multigrid methods for solving more difficult problems …
The solution of parameter-dependent linear systems, by classical methods, leads to an arithmetic effort that grows exponentially in the number of parameters. This renders the multigrid method, which has a well understood convergence theory, infeasible. A parameter-dependent representation, e.g., a low-rank tensor format, can avoid this exponential dependence, but in these it is unknown how to calculate the inverse directly within the representation. The combination of these representations with the multigrid method requires a parameter-dependent version of the classical multigrid theory and a …
Numerical simulation of incompressible viscous flow, in particular in three space dimensions, continues to remain a challenging task. Space-time finite element methods feature the natural construction of higher order discretization schemes. They offer the potential to achieve accurate results on computationally feasible grids. Linearizing the resulting algebraic problems by Newton's method yields linear systems with block matrices built of $(k+1)\times (k+1)$ saddle point systems, where $k$ denotes the polynomial order of the variational time discretization. We demonstrate numerically the effi…
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 $2k$…
We consider an additive Vanka-type smoother for the Poisson equation discretized by the standard finite difference centered scheme. Using local Fourier analysis, we derive analytical formulas for the optimal smoothing factors for two types of smoothers, called vertex-wise and element-wise Vanka smoothers, and present the corresponding stencils. Interestingly, in one dimension the element-wise Vanka smoother is equivalent to the scaled mass operator obtained from the linear finite element method, and in two dimensions the element-wise Vanka smoother is equivalent to the scaled mass operator dis…
Emory University（埃默里大学）；Lawrence Livermore National Laboratory（劳伦斯利弗莫尔国家实验室）
Multigrid methods are one of the most efficient techniques for solving linear systems arising from Partial Differential Equations (PDEs) and graph Laplacians from machine learning applications. One of the key components of multigrid is smoothing, which aims at reducing high-frequency errors on each grid level. However, finding optimal smoothing algorithms is problem-dependent and can impose challenges for many problems. In this paper, we propose an efficient adaptive framework for learning optimized smoothers from operator stencils in the form of convolutional neural networks (CNNs). The CNNs …
This paper presents a numerical study on multigrid algorithms of $V$-cycle type for problems posed in the Hilbert space $H(\mathbf{curl})$ in three dimensions. The multigrid methods are designed for discrete problems originated from the discretization using the hexahedral Nédélec edge element of the lowest-order. Our suggested methods are associated with smoothers constructed by substructuring based on domain decomposition methods of nonoverlapping type. Numerical experiments to demonstrate the robustness and the effectiveness of the suggested algorithms are also provided.
Two-level convergence of Algebraic Multigrid with Overlapping Smoothers and Spectral Coarse Grids
Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室）；Science and Technology Facilities Council, Scientific Computing Department, Rutherford Appleton Laboratory（英国科学与技术设施委员会，科学计算部，卢瑟福阿普尔顿实验室）
On the analysis of block smoothers for saddle point problems
Institute of System Simulation, University Erlangen-Nuremberg（埃尔朗根-纽伦堡大学系统模拟研究所）；Institute of Computational Mathematics, Johannes Kepler University Linz（林茨约翰内斯开普勒大学计算数学研究所）
A comparative study of efficient multigrid solvers for high-order local discontinuous Galerkin methods: Poisson, elliptic interface, and multiphase Stokes problems
Neumann series underlie both Krylov methods and algebraic multigrid smoothers. A low-synch modified Gram-Schmidt (MGS)-GMRES algorithm is described that employs a Neumann series to accelerate the projection step. A corollary to the backward stability result of Paige et al. (2006) demonstrates that the truncated Neumann series approximation is sufficient for convergence of GMRES. The lower triangular solver associated with the correction matrix $T_m = (\: I + L_m \:)^{-1}$ may then be replaced by a matrix-vector product with $T_m = I - L_m$. Next, Neumann series are applied to accelerate the cl…
During the last decade, Neural Networks (NNs) have proved to be extremely effective tools in many fields of engineering, including autonomous vehicles, medical diagnosis and search engines, and even in art creation. Indeed, NNs often decisively outperform traditional algorithms. One area that is only recently attracting significant interest is using NNs for designing numerical solvers, particularly for discretized partial differential equations. Several recent papers have considered employing NNs for developing multigrid methods, which are a leading computational tool for solving discretized p…
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.
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…
In this work, we propose three Braess-Sarazin-type multigrid relaxation schemes for solving linear elasticity problems, where the marker and cell scheme, a finite difference method, is used for the discretization. The three relaxation schemes are Jacobi-Braess-Sarazin, Mass-Braess-Sarazin, and Vanka-Braess-Sarazin. A local Fourier analysis (LFA) for the block-structured relaxation schemes is presented to study multigrid convergence behavior. From LFA, we derive optimal LFA smoothing factor for each case. We obtain highly efficient smoothing factors, which are independent of Lamé constants. Van…
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.
In this paper we propose and analyze new efficient sparse approximate inverse (SPAI) smoothers for solving the two-dimensional (2D) and three-dimensional (3D) Laplacian linear system with geometric multigrid methods. Local Fourier analysis shows that our proposed SPAI smoother for 2D achieves a much smaller smoothing factor than the state-of-the-art SPAI smoother studied in [Bolten, M., Huckle, T.K. and Kravvaritis, C.D., 2016. Sparse matrix approximations for multigrid methods. Linear Algebra and its Applications, 502, pp.58-76.]. The proposed SPAI smoother for 3D cases provides smaller optim…
Scalable smoothing strategies for a geometric multigrid method for the immersed boundary equations
University of North Carolina, Chapel Hill（北卡罗来纳大学教堂山分校）；Rice University（莱斯大学）；Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）；University of California, Davis（加州大学戴维斯分校）
We consider the numerical solution of time-dependent space tempered fractional diffusion equations. The use of Crank-Nicolson in time and of second-order accurate tempered weighted and shifted Grünwald difference in space leads to dense (multilevel) Toeplitz-like linear systems. By exploiting the related structure, we design an ad-hoc multigrid solver and multigrid-based preconditioners, all with weighted Jacobi as smoother. A new smoothing analysis is provided, which refines state-of-the-art results expanding the set of the suitable Jacobi weights. Furthermore, we prove that if a multigrid me…
We present families of space-time finite element methods (STFEMs) for a coupled hyperbolic-parabolic system of poro- or thermoelasticity. Well-posedness of the discrete problems is proved. Higher order approximations inheriting most of the rich structure of solutions to the continuous problem on computationally feasible grids are naturally embedded. However, the block structure and solution of the algebraic systems become increasingly complex for these members of the families. We present and analyze a robust geometric multigrid (GMG) preconditioner for GMRES iterations. The GMG method uses a l…
We present a parallelized geometric multigrid (GMG) method, based on the cell-based Vanka smoother, for higher order space-time finite element methods (STFEM) to the incompressible Navier--Stokes equations. The STFEM is implemented as a time marching scheme. The GMG solver is applied as a preconditioner for GMRES iterations. Its performance properties are demonstrated for 2d and 3d benchmarks of flow around a cylinder. The key ingredients of the GMG approach are the construction of the local Vanka smoother over all degrees of freedom in time of the respective subinterval and its efficient appl…
Comparison of Multigrid Algorithms for High-order Continuous Finite Element Discretizations
School of Computing, University of Utah（犹他大学计算学院）；Courant Institute of Mathematical Sciences, New York University（纽约大学柯朗数学科学研究所）；Institute for Computational Engineering and Sciences, The University of Texas at Austin（德克萨斯大学奥斯汀分校计算工程与科学研究所）
Multigrid methods for Hdiv-conforming discontinuous Galerkin methods for the Stokes equations
Interdisziplinäres Zentrum für Wissenschaftliches Rechnen (IWR), Universität Heidelberg（海德堡大学计算科学跨学科中心）；Department of Mathematics, Texas A&M University（德克萨斯农工大学数学系）
