HSL_MI20 : An efficient AMG preconditioner for finite element problems in 3D (original) (raw)
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HSL_MI20 : an efficient AMG preconditioner
2007
Algebraic multigrid (AMG) is an efficient multilevel method for solving large sparse linear systems obtained from the discretisation of scalar eliptic partial differential equations. AMG can be used to compute powerful preconditioners for use with Krylov subspace methods. We report on the design and development of an efficient, robust and portable implementation of AMG that is available as package HSL_MI20 within the HSL mathematical software library. HSL_MI20 implements the classical (Ruge-Stuben) AMG method and, although it can be used as a "black-box" preconditioner, it offers the user a large number of options and parameters that may be tuned to enhance its performance for specific applications. The performance of HSL_MI20 is illustrated using finite element discretisations of diffusion and convection-diffusion problems in three dimensions
An efficient algebraic multigrid preconditioned conjugate gradient solver
Computer Methods in Applied Mechanics and Engineering, 2003
In this paper, we present a robust and efficient algebraic multigrid preconditioned conjugate gradient solver for systems of linear equations arising from the finite element discretization of a scalar elliptic partial differential equation of second order on unstructured meshes. The algebraic multigrid (AMG) method is one of most promising methods for solving large systems of linear equations arising from unstructured meshes. The conventional AMG method usually requires an expensive setup time, particularly for three dimensional problems so that generally it is not used for small and medium size systems or low-accuracy approximations. Our solver has a quick setup phase for the AMG method and a fast iteration cycle. These allow us to apply this solver for not only large systems but also small to medium systems of linear equations and also for systems requiring low-accuracy approximations.
International Journal for Numerical Methods in Engineering, 2010
A new library called FLEXMG has been developed for a spectral/finite-element incompressible flow solver called SFELES. FLEXMG allows to use various types of iterative solvers preconditioned by algebraic multigrid methods. Two families of algebraic multigrid preconditioners have been implemented, of smooth aggregation-type and non nested finite-element-type. Unlike gridless multigrid, both of these families use the information contained in the initial fine mesh. Our aggregation-type multigrid is smoothed with either a constant or a linear least square fitting function while the non nested finite-element-type multigrid is already smooth by construction. All these multigrid preconditioners are tested as stand-alone solvers or coupled to a GMRES method. After analyzing the accuracy of our solvers on a typical test case in fluid mechanics, their performance in terms of convergence rate, computational speed and memory consumption are compared with the performance of a direct sparse LU solver as a reference. Finally, the importance of using smooth interpolation operators is also underlined in the study.
Uniciencia, 2020
This paper presents detailed aspects regarding the implementation of the Finite Element Method (FEM) to solve a Poisson’s equation with homogeneous boundary conditions. The aim of this paper is to clarify details of this implementation, such as the construction of algorithms, implementation of numerical experiments, and their results. For such purpose, the continuous problem is described, and a classical FEM approach is used to solve it. In addition, a multilevel technique is implemented for an efficient resolution of the corresponding linear system, describing and including some diagrams to explain the process and presenting the implementation codes in MATLAB®. Finally, codes are validated using several numerical experiments. Results show an adequate behavior of the preconditioner since the number of iterations of the PCG method does not increase, even when the mesh size is reduced.
MultiGrid Preconditioners for Mixed Finite Element Methods of the Vector Laplacian
Journal of Scientific Computing, 2018
Due to the indefiniteness and poor spectral properties, the discretized linear algebraic system of the vector Laplacian by mixed finite element methods is hard to solve. A block diagonal preconditioner has been developed and shown to be an effective preconditioner by Arnold et al. (Acta Numer 15:1–155, 2006). The purpose of this paper is to propose alternative and effective block diagonal and approximate block factorization preconditioners for solving these saddle point systems. A variable V-cycle multigrid method with the standard point-wise Gauss–Seidel smoother is proved to be a good preconditioner for the discrete vector Laplacian operator. The major benefit of our approach is that the point-wise Gauss–Seidel smoother is more algebraic and can be easily implemented as a black-box smoother. This multigrid solver will be further used to build preconditioners for the saddle point systems of the vector Laplacian. Furthermore it is shown that Maxwell’s equations with the divergent fr...
2007
Application of algebraic multigrid method and approximate sparse inverses are applied as preconditioners for large algebraic systems arising in approximation of diffusion-reaction problems in 3-dimensional complex domains. Here we report the results of numerical experiments when using highly graded and locally refined meshes for problems with non-homogeneous and anisotropic coefficients that have small features and almost singular solutions. For the discretization of the domain and the finite element approximation we have used the system AGGIEFEM, a universal computational tool for PDEs developed in the VIGRE seminar in Introduction to Scientific Computing at TAMU. For solving the algebraic system we have used ParaSails and BoomerAMG preconditioners that are part of the HYPRE (High Performance Preconditioners) library developed in CASC at Lawrence Livermore National Laboratory. *
Geophysical Journal International, 2014
We present an elaborate preconditioning scheme for Krylov subspace methods which has been developed to improve the performance and reduce the execution time of parallel node-based finite-element (FE) solvers for 3-D electromagnetic (EM) numerical modelling in exploration geophysics. This new preconditioner is based on algebraic multigrid (AMG) that uses different basic relaxation methods, such as Jacobi, symmetric successive over-relaxation (SSOR) and Gauss-Seidel, as smoothers and the wave front algorithm to create groups, which are used for a coarse-level generation. We have implemented and tested this new preconditioner within our parallel nodal FE solver for 3-D forward problems in EM induction geophysics. We have performed series of experiments for several models with different conductivity structures and characteristics to test the performance of our AMG preconditioning technique when combined with biconjugate gradient stabilized method. The results have shown that, the more challenging the problem is in terms of conductivity contrasts, ratio between the sizes of grid elements and/or frequency, the more benefit is obtained by using this preconditioner. Compared to other preconditioning schemes, such as diagonal, SSOR and truncated approximate inverse, the AMG preconditioner greatly improves the convergence of the iterative solver for all tested models. Also, when it comes to cases in which other preconditioners succeed to converge to a desired precision, AMG is able to considerably reduce the total execution time of the forward-problem code-up to an order of magnitude. Furthermore, the tests have confirmed that our AMG scheme ensures grid-independent rate of convergence, as well as improvement in convergence regardless of how big local mesh refinements are. In addition, AMG is designed to be a black-box preconditioner, which makes it easy to use and combine with different iterative methods. Finally, it has proved to be very practical and efficient in the parallel context.
A Block-Diagonal Algebraic Multigrid Preconditioner for the Brinkman Problem
The Brinkman model is a unified law governing the flow of a viscous fluid in cavity (Stokes equations) and in porous media (Darcy equations). In this work, we explore a novel mixed formulation of the Brinkman problem by introducing the flow's vorticity as an additional unknown. This formulation allows for a uniformly stable and conforming discretization by standard finite element (Nédélec, Raviart-Thomas, discontinuous piecewise polynomials). Based on the stability analysis of the problem in the H(curl) − H(div) − L 2 norms ([24]), we study a scalable block diagonal preconditioner which is provably optimal in the constant coefficient case. Such preconditioner takes advantage of the parallel auxiliary space AMG solvers for H(curl) and H(div) problems available in hypre ([11]). The theoretical results are illustrated by numerical experiments.
The Mixed Finite Element Multigrid Preconditioned MINRES Method for Stokes Equations
Journal of Applied Fluid Mechanics
The study considers the saddle point problem arising from the mixed finite element discretization of the steady state Stokes equations. The saddle point problem is an indefinite system of linear equations, a feature that degrades the performance of any iterative solver. The heart of the study is the construction of fast, robust and effective iterative solution methods for such systems. Specific attention is given to the preconditioned MINRES solver PMINRES which is carefully treated for the solution of the Stokes equations. The study concentrates on the block preconditioner applied to the MINRES to effectively solve the whole coupled system. We combine iterative techniques with the MINRES as preconditioner approximations to produce an efficient solver for indefinite system of equations. We consider different preconditioner approximations of the building blocks of the preconditioner and compare their effects in accelerating the MINRES iterative scheme. We give a detailed overview of the algorithmic aspects and the theoretical convergence analysis of our solver. We study the MINRES method with the following preconditioner approximations: diagonal, multigrid v-cycle, preconditioned conjugate gradient and Chebyshev semi iteration methods. A comparative analysis of the preconditioner approximations show that the multigrid method is a suitable accelerator for the MINRES method. The application of the preconditioner becomes mandatory as evidenced by poor performance of the MINRES as compared to PMINRES. We study the problem in a two dimensional setting using the Hood-Taylor Q 2 − Q 1 stable pair of finite elements. The incompressible flow iterative solution software(IFISS) matlab toolbox is used to assemble the matrices. We present the numerical results to illustrate the efficiency and robustness of the MINRES scheme with the multigrid preconditioner.
Preconditioning and Two-Level Multigrid Methods of Arbitrary Degree of Approximation
Mathematics of Computation, 1983
Let A be a mesh parameter corresponding to a finite element mesh for an elliptic problem. We describe preconditioning methods for two-level meshes which, for most problems solved in practice, behave as methods of optimal order in both storage and computational complexity. Namely, per mesh point, these numbers are bounded above by relatively small constants for all h > h0, where h0 is small enough to cover all but excessively fine meshes. We note that, in practice, multigrid methods are actually solved on a finite, often even a fixed number of grid levels, in which case also these methods are not asymptotically optimal as h-> 0. Numerical tests indicate that the new methods are about as fast as the best implementations of multigrid methods applied on general problems (variable coefficients, general domains and boundary conditions) for all but excessively fine meshes. Furthermore, most of the latter methods have been implemented only for difference schemes of second order of accuracy, whereas our methods are applicable to higher order approximations. We claim that our scheme could be added fairly easily to many existing finite element codes.