Nonuniformly weighted Schwarz smoothers for spectral element multigrid

A hybrid Schwarz/multigrid method for spectral element solvers to the Poisson equation in $\mathbb R^2$ is presented. It extends the additive Schwarz method studied by J. Lottes and P. Fischer (J. Sci. Comput. 24:45--78, 2005) by introducing nonuniform weight distributions based on the smoothed sign function. Using a V-cycle with only one pre-smoothing, the new method attains logarithmic convergence rates in the range from 1.2 to 1.9, which corresponds to residual reductions of almost two orders of magnitude. Compared to the original method, it reduces the iteration count by a factor of 1.5 to 3, leading to runtime savings of about 50 percent. In numerical experiments the method proved robust with respect to the mesh size and polynomial orders up to 32. Used as a preconditioner for the (inexact) CG method it is also suited for anisotropic meshes and easily extended to diffusion problems with variable coefficients.Comment: Multigrid method; Schwarz methods; spectral element method; p-version finite element metho

Development of Krylov and AMG linear solvers for large-scale sparse matrices on GPUs

This research introduce our work on developing Krylov subspace and AMG solvers on NVIDIA GPUs. As SpMV is a crucial part for these iterative methods, SpMV algorithms for single GPU and multiple GPUs are implemented. A HEC matrix format and a communication mechanism are established. And also, a set of specific algorithms for solving preconditioned systems in parallel environments are designed, including ILU(k), RAS and parallel triangular solvers. Based on these work, several Krylov solvers and AMG solvers are developed. According to numerical experiments, favorable acceleration performance is acquired from our Krylov solver and AMG solver under various parameter conditions

Recent Results on Domain Decomposition Preconditioning for the High-frequency Helmholtz Equation using Absorption

In this paper we present an overview of recent progress on the development and analysis of domain decomposition preconditioners for discretised Helmholtz problems, where the preconditioner is constructed from the corresponding problem with added absorption. Our preconditioners incorporate local subproblems that can have various boundary conditions, and include the possibility of a global coarse mesh. While the rigorous analysis describes preconditioners for the Helmholtz problem with added absorption, this theory also informs the development of efficient multilevel solvers for the "pure" Helmholtz problem without absorption. For this case, 2D experiments for problems containing up to about $50$ wavelengths are presented. The experiments show iteration counts of order about $\mathcal{O}(n^{0.2})$ and times (on a serial machine) of order about $\mathcal{O}(n^{\alpha})$, { with $\alpha \in [1.3,1.4]$} for solving systems of dimension $n$. This holds both in the pollution-free case corresponding to meshes with grid size $\mathcal{O}(k^{-3/2})$ (as the wavenumber $k$ increases), and also for discretisations with a fixed number of grid points per wavelength, commonly used in applications. Parallelisation of the algorithms is also briefly discussed.Comment: To appear in: Modern Solvers for Helmholtz problems edited by D. Lahaye, J. Tang and C. Vuik, Springer Geosystems Mathematics series, 201

A Multilevel Approach for Trace System in HDG Discretizations

We propose a multilevel approach for trace systems resulting from hybridized discontinuous Galerkin (HDG) methods. The key is to blend ideas from nested dissection, domain decomposition, and high-order characteristic of HDG discretizations. Specifically, we first create a coarse solver by eliminating and/or limiting the front growth in nested dissection. This is accomplished by projecting the trace data into a sequence of same or high-order polynomials on a set of increasingly $h-$coarser edges/faces. We then combine the coarse solver with a block-Jacobi fine scale solver to form a two-level solver/preconditioner. Numerical experiments indicate that the performance of the resulting two-level solver/preconditioner depends only on the smoothness of the solution and is independent of the nature of the PDE under consideration. While the proposed algorithms are developed within the HDG framework, they are applicable to other hybrid(ized) high-order finite element methods. Moreover, we show that our multilevel algorithms can be interpreted as a multigrid method with specific intergrid transfer and smoothing operators. With several numerical examples from Poisson, pure transport, and convection-diffusion equations we demonstrate the robustness and scalability of the algorithms

Robust multigrid for high-order discontinuous Galerkin methods: A fast Poisson solver suitable for high-aspect ratio Cartesian grids

We present a polynomial multigrid method for nodal interior penalty and local discontinuous Galerkin formulations of the Poisson equation on Cartesian grids. For smoothing we propose two classes of overlapping Schwarz methods. The first class comprises element-centered and the second face-centered methods. Within both classes we identify methods that achieve superior convergence rates, prove robust with respect to the mesh spacing and the polynomial order, at least up to ${P=32}$. Consequent structure exploitation yields a computational complexity of $O(PN)$, where $N$ is the number of unknowns. Further we demonstrate the suitability of the face-centered method for element aspect ratios up to 32

Multigrid algorithms for hphp-Discontinuous Galerkin discretizations of elliptic problems

We present W-cycle multigrid algorithms for the solution of the linear system of equations arising from a wide class of $hp$-version discontinuous Galerkin discretizations of elliptic problems. Starting from a classical framework in multigrid analysis, we define a smoothing and an approximation property, which are used to prove the uniform convergence of the W-cycle scheme with respect to the granularity of the grid and the number of levels. The dependence of the convergence rate on the polynomial approximation degree $p$ is also tracked, showing that the contraction factor of the scheme deteriorates with increasing $p$. A discussion on the effects of employing inherited or non-inherited sublevel solvers is also presented. Numerical experiments confirm the theoretical results

A highly parallel multilevel Newton-Krylov-Schwarz method with subspace-based coarsening and partition-based balancing for the multigroup neutron transport equations on 3D unstructured meshes

The multigroup neutron transport equations have been widely used to study the motion of neutrons and their interactions with the background materials. Numerical simulation of the multigroup neutron transport equations is computationally challenging because the equations is defined on a high dimensional phase space (1D in energy, 2D in angle, and 3D in spatial space), and furthermore, for realistic applications, the computational spatial domain is complex and the materials are heterogeneous. The multilevel domain decomposition methods is one of the most popular algorithms for solving the multigroup neutron transport equations, but the construction of coarse spaces is expensive and often not strongly scalable when the number of processor cores is large. In this paper, we study a highly parallel multilevel Newton-Krylov-Schwarz method equipped with several novel components, such as subspace-based coarsening, partition-based balancing and hierarchical mesh partitioning, that enable the overall simulation strongly scalable in terms of the compute time. Compared with the traditional coarsening method, the subspace-based coarsening algorithm significantly reduces the cost of the preconditioner setup that is often unscalable. In addition, the partition-based balancing strategy enhances the parallel efficiency of the overall solver by assigning a nearly-equal amount of work to each processor core. The hierarchical mesh partitioning is able to generate a large number of subdomains and meanwhile minimizes the off-node communication. We numerically show that the proposed algorithm is scalable with more than 10,000 processor cores for a realistic application with a few billions unknowns on 3D unstructured meshes.Comment: Submitted to SIAM Journal on Scientific Computing. 25 pages and 9 figure

V-cycle multigrid algorithms for discontinuous Galerkin methods on non-nested polytopic meshes

In this paper we analyse the convergence properties of V-cycle multigrid algorithms for the numerical solution of the linear system of equations arising from discontinuous Galerkin discretization of second-order elliptic partial differential equations on polytopal meshes. Here, the sequence of spaces that stands at the basis of the multigrid scheme is possibly non nested and is obtained based on employing agglomeration with possible edge/face coarsening. We prove that the method converges uniformly with respect to the granularity of the grid and the polynomial approximation degree p, provided that the number of smoothing steps, which depends on p, is chosen sufficiently large.Comment: 26 pages, 23 figures, submitted to Journal of Scientific Computin

Coarse spaces over the ages

The objective of this paper is to explain the principles of the design of a coarse space in a simplified way and by pictures. The focus is on ideas rather than on a more historically complete presentation. Also, space limitation does not allow even to mention many important methods and papers that should be rightfully included.Comment: Minor changes. 8 pages, 4 figures, to appear in DD19 proceeding

Multigrid Methods for Hellan-Herrmann-Johnson Mixed Method of Kirchhoff Plate Bending Problems

A V-cycle multigrid method for the Hellan-Herrmann-Johnson (HHJ) discretization of the Kirchhoff plate bending problems is developed in this paper. It is shown that the contraction number of the V-cycle multigrid HHJ mixed method is bounded away from one uniformly with respect to the mesh size. The uniform convergence is achieved for the V-cycle multigrid method with only one smoothing step and without full elliptic regularity. The key is a stable decomposition of the kernel space which is derived from an exact sequence of the HHJ mixed method, and the strengthened Cauchy Schwarz inequality. Some numerical experiments are provided to confirm the proposed V-cycle multigrid method. The exact sequences of the HHJ mixed method and the corresponding commutative diagram is of some interest independent of the current context.Comment: 23 page