Algebraic Multigrid for Linear Systems Obtained by Explicit Element Reduction

Brunner, Thomas A.; Kolev, Tzanio V.

doi:10.1137/100801640

Cited by 5 publications

(4 citation statements)

References 35 publications

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“…The purpose of this section, which can be read independently of the rest of the paper, is to present a simple matrix result, whose relevance to our problem will be clear in the next section. The result is a generalization of [7,Lemma 4.2]. Suppose i∪f = {1, 2, .…”

Section: General Meshesmentioning

confidence: 80%

A scalable preconditioner for a DPG method

Barker,

Dobrev,

Gopalakrishnan

et al. 2016

Preprint

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We show how a scalable preconditioner for the primal discontinuous Petrov-Galerkin (DPG) method can be developed using existing algebraic multigrid (AMG) preconditioning techniques. The stability of the DPG method gives a norm equivalence which allows us to exploit existing AMG algorithms and software. We show how these algebraic preconditioners can be applied directly to a Schur complement system of interface unknowns arising from the DPG method. To the best of our knowledge, this is the first massively scalable algebraic preconditioner for DPG problems.

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Section: General Meshesmentioning

confidence: 80%

A scalable preconditioner for a DPG method

Barker,

Dobrev,

Gopalakrishnan

et al. 2016

Preprint

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“…In this work, we focus on algebraic multigrid (AMG) methods: these methods give essentially black-box highly scalable preconditioners for A h requiring minimal discretization information. AMG convergence for lowest-order H 1 and DG finite element discretizations for elliptic problems has been extensively studied in the literature [14,33,57,70] and has further been extended to definite H(curl) [15,48] and H(div) [30,49] problems. Additionally, several high-performance massively parallel and GPU-accelerated implementations such as the BoomerAMG, AMS and ADS preconditioners in the hypre library [34] are available.…”

Section: Algebraic Multigrid Preconditioningmentioning

confidence: 99%

Low-order preconditioning for the high-order finite element de Rham complex

Pazner¹,

Kolev²,

Dohrmann³

2022

Preprint

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In this paper we present a unified framework for constructing spectrally equivalent low-orderrefined discretizations for the high-order finite element de Rham complex. This theory covers diffusion problems in H 1 , H(curl), and H(div), and is based on combining a low-order discretization posed on a refined mesh with a high-order basis for Nédélec and Raviart-Thomas elements that makes use of the concept of polynomial histopolation (polynomial fitting using prescribed mean values over certain regions). This spectral equivalence, coupled with algebraic multigrid methods constructed using the low-order discretization, results in highly scalable matrix-free preconditioners for high-order finite element problems in the full de Rham complex. Additionally, a new lowest-order (piecewise constant) preconditioner is developed for highorder interior penalty discontinuous Galerkin (DG) discretizations, for which spectral equivalence results and convergence proofs for algebraic multigrid methods are provided. In all cases, the spectral equivalence results are independent of polynomial degree and mesh size; for DG methods, they are also independent of the penalty parameter. These new solvers are flexible and easy to use; any "black-box" preconditioner for low-order problems can be used to create an effective and efficient preconditioner for the corresponding high-order problem. A number of numerical experiments are presented, using the finite element library MFEM, with which the construction of such preconditioners requires only one or two lines of code. The theoretical properties of these preconditioners are corroborated, and the flexibility and scalability of the method are demonstrated on a range of challenging three-dimensional problems.

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“…Of course, for efficiency, we would like to build a preconditioner directly for the (sparse) matrix S and not for the original matrix A. Such preconditioners are discussed in [10] where it is shown, both theoretically and practically, that BoomerAMG and AMS work well on Schur complements of H 1 and H(curl) discretizations respectively. Similar analysis for the H(div) case was recently performed in [5].…”

Section: Scalable Algebraic H(div) Preconditioningmentioning

confidence: 99%

“…Thus, these two approaches are of great practical interest and the goal of the present paper is to study them in a common framework, emphasize their similarities and differences, and most importantly, design new, or modify existing solution techniques that are efficient and scalable, achieving substantial savings compared to the state-of-the-art solvers for the original problems. For problems obtained by static condensation, there has been success in modifying the state-of-the-art solvers (AMS and ADS) to be directly applicable to the reduced problem, cf., [10,5]. In this paper, we focus on the design of scalable solvers for problems involving H(div) in a general algebraic setting, extending the preliminary results in [25].…”

mentioning

confidence: 98%

Algebraic Hybridization and Static Condensation with Application to Scalable $H$(div) Preconditioning

Dobrev¹,

Kolev²,

Lee³

et al. 2019

SIAM J. Sci. Comput.

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We propose an unified algebraic approach for static condensation and hybridization, two popular techniques in finite element discretizations. The algebraic approach is supported by the construction of scalable solvers for problems involving H(div)-spaces discretized by conforming (Raviart-Thomas) elements of arbitrary order. We illustrate through numerical experiments the relative performance of the two (in some sense dual) techniques in comparison with a state-of-the-art parallel solver, ADS [24], available in [1] and [2]. The superior performance of the hybridization technique is clearly demonstrated with increased benefit for higher order elements.1. Introduction. This paper proposes a unified algebraic approach of constructing reduced problems by two popular techniques in the finite element method: static condensation and hybridization. It also discusses approaches for constructing scalable preconditioners for these reduced problems. The present work is motivated by the solution of problems arising in variety of finite element simulations and of particular interest is the case when the problems involve the function space H(div) (L 2 -vector functions that have their weak divergence in L 2 ). Such problems naturally arise in the mixed finite element method for Darcy equation [6], Brinkman equation [27], as well as in the FOSLS (first order system least-squares) discretization approach, [11]. Also, certain formulations of radiation-diffusion transport [9], lead to problems involving the space H(div). This is in fact one of the practical examples from a more realistic simulation that we consider in our numerical tests.Highly scalable solvers for H(div) problems have been designed in the past. The most successful ones, of ADS-type [24], were based on the regular decomposition theory developed by Hiptmair and Xu [20]. Although of optimal complexity, the current state-of-the-art ADS algorithms are quite involved and require also going through solvers for H(curl), such as AMS, [23]. Both solvers, ADS and AMS, require some additional user input (e.g., discrete gradient), which makes them not completely algebraic. The results of the present paper can be viewed as a further step towards the design of fully algebraic solvers for problems involving H(div).To design faster, yet still scalable, alternative to ADS, we employ a traditional finite element technique commonly used in mixed finite elements referred to as hybridization. This is perhaps the first technique proposed to solve the saddle-point problems arising in the mixed methods, since it leads to symmetric and positive definite (SPD) reduced problems. Its early form appeared in a paper by Fraeijis de Veubekein in 1965 [17] as an efficient solution strategy for elasticity problems. In [3], Arnold and Brezzi studied the method from a theoretical perspective, and obtained *

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Algebraic Multigrid for Linear Systems Obtained by Explicit Element Reduction

Cited by 5 publications

References 35 publications

A scalable preconditioner for a DPG method

A scalable preconditioner for a DPG method

Low-order preconditioning for the high-order finite element de Rham complex

Algebraic Hybridization and Static Condensation with Application to Scalable $H$(div) Preconditioning

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