Overlapping Schwarz methods in H(curl) on polyhedral domains

Pasciak, Joseph E.; Zhao, Jun

doi:10.1515/jnma.2002.221

Cited by 51 publications

(44 citation statements)

References 24 publications

(38 reference statements)

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“…There are other decompositions of H 0 (curl; Ω) ∩ H(div 0 ; Ω) that can be exploited for the purpose of preconditioning H(curl; Ω) conforming methods [22,17].…”

Section: Corollary 25 We Have ∇Hmentioning

confidence: 99%

Hodge decomposition for divergence-free vector fields and two-dimensional Maxwell’s equations

et al. 2011

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“…There are other decompositions of H 0 (curl; Ω) ∩ H(div 0 ; Ω) that can be exploited for the purpose of preconditioning H(curl; Ω) conforming methods [22,17].…”

Section: Corollary 25 We Have ∇Hmentioning

confidence: 99%

Hodge decomposition for divergence-free vector fields and two-dimensional Maxwell’s equations

et al. 2011

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“…Of special note are the recent papers [23,28,29,30] which uses the idea of auxiliary space preconditioners and nodal vector Laplacians. The approach in this paper bears some similarity to our proposed method; however, it is based on a discrete version of a non-orthogonal "regular decomposition" [34] of H(curl), as opposed to a discrete Hodge decomposition in our case.…”

mentioning

confidence: 99%

An Algebraic Multigrid Approach Based on a Compatible Gauge Reformulation of Maxwell's Equations

Bochev¹,

Hu²,

Siefert³

et al. 2008

SIAM J. Sci. Comput.

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Abstract. With the rise in popularity of compatible finite element, finite difference and finite volume discretizations for the time domain eddy current equations, there has been a corresponding need for fast solvers of the resulting linear algebraic systems. However, the traits that make compatible discretizations a preferred choice for the Maxwell's equations also render these linear systems essentially intractable by truly black-box techniques. We propose a new algebraic reformulation of the discrete eddy current equations along with a new algebraic multigrid technique (AMG) for this reformulated problem. The reformulation process takes advantage of a discrete Hodge decomposition on co-chains to replace the discrete eddy current equations by an equivalent 2 × 2 block linear system whose diagonal blocks are discrete Hodge Laplace operators acting on 1-cochains and 0-cochains, respectively. While this new AMG technique requires somewhat specialized treatment on the finest mesh, the coarser meshes can be handled using standard methods for Laplace-type problems. Our new AMG method is applicable to a wide range of compatible methods on structured and unstructured grids, including edge finite elements, mimetic finite differences, co-volume methods and Yee-like schemes. We illustrate the new technique, using edge elements in the context of smoothed aggregation AMG, and present computational results for problems in both two and three dimensions.Key words. Maxwell's equations, eddy currents, algebraic multigrid, multigrid, discrete Hodge Laplacian, compatible discretizations, edge elements. AMS subject classifications. 65F10, 65F30, 78A301. Introduction. Due to the pioneering work of Bossavit [9], it is now wellknown that stable and accurate numerical solution of Maxwell's equations can be achieved by using discrete spaces from a finite-dimensional analogue of the differential De Rham complex. Numerical methods that are based on such discretizations are now commonly referred to as compatible, or mimetic methods [6].However, the same traits that make compatible methods a natural choice for Maxwell's equations also render the ensuing linear system essentially intractable by truly general purpose black-box multigrid solvers. For example, a compatible discretization of the curl-curl operator gives rise to a symmetric, semi-definite linear system whose null-space is of approximately the same size as the number of nodes in the computational grid. Multilevel solution of such systems often utilizes special smoothing, prolongation and restriction operators that separate error components in the null-space and its complement and satisfy a commuting diagram property. Formulation of such operators is well-understood in geometric multigrid settings [22], where

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“…This robustness is based on the following fundamental theoretical result by Hiptmair and Xu [19], see also [30], which provides a general statement about the structure of the space V h , similar to the classical Helmholtz decomposition [28, §7.2.1]. We use the notation f g and f g to denote that f ≤ Cg and f ≥ Cg, respectively with an absolute constant C.…”

Section: Auxiliary-space Maxwell Solver (Ams)mentioning

confidence: 99%

Algebraic Multigrid for Linear Systems Obtained by Explicit Element Reduction

Brunner¹,

Kolev²

2011

SIAM J. Sci. Comput.

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Abstract. We consider sparse linear systems, where a set of "interior" degrees of freedom have been eliminated in order to reduce the problem size. This elimination is assumed to be local, so the "interior" principal sub-matrix is block-diagonal, and the resulting Schur complement is still sparse. For it to be beneficial, the elimination process should lead to reduced memory requirements, but equally importantly, it should also result in an algebraic problem that can be solved efficiently. In this paper we propose a general element reduction approach, and show how the elimination process can exploit a particular "sub-zonal" discretization to maintain the sparsity of the Schur complement. We also investigate Algebraic Multigrid (AMG) solution algorithms applied to the reduced problem, and we discuss the influence of the local elimination on solver-related properties of the matrix, such as near-nullspace preservation and the availability of stable subspace decompositions. We focus on BoomerAMG, a parallel variant of classical Ruge-Stüben AMG, applied to scalar diffusion problems [18], and the Auxiliary-space Maxwell Solver (AMS) for electromagnetic diffusion applications [22]. In the electromagnetic case, we establish algebraically a reduced version of the HX decomposition from [19], and consider a modification of the reduction process that targets the singular problems arising in simulations with pure void (zero conductivity) regions. For scalar diffusion problems, our particular stencil analysis shows that the reduction has a positive effect on meshes with stretched elements. We present a number of 2D, 3D and axi-symmetric numerical experiments, which demonstrate that the combination of an appropriately chosen local elimination with the use of the BoomerAMG and AMS solvers can lead to significant improvements in the overall solution time.

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Overlapping Schwarz methods in H(curl) on polyhedral domains

Cited by 51 publications

References 24 publications

Hodge decomposition for divergence-free vector fields and two-dimensional Maxwell’s equations

Hodge decomposition for divergence-free vector fields and two-dimensional Maxwell’s equations

An Algebraic Multigrid Approach Based on a Compatible Gauge Reformulation of Maxwell's Equations

Algebraic Multigrid for Linear Systems Obtained by Explicit Element Reduction

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