A Moment-Parity Multigrid Preconditioner for the First-Order System Least-Squares Formulation of the Boltzmann Transport Equation

This paper describes a preconditioned conjugate gradient scheme for the Pn-h finite element discretization of a first-order system least-squares (FOSLS) formulation of the Boltzmann transport equation. The preconditioner is based on the norm equivalence between the FOSLS functional and a V norm. Its...

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Bibliographic Details
Published in:SIAM journal on scientific computing Vol. 25; no. 2; pp. 513 - 533
Main Authors: Brown, P. N., Lee, B., Manteuffel, T. A.
Format: Journal Article
Language:English
Published: Philadelphia Society for Industrial and Applied Mathematics 01.01.2003
Subjects:
Algorithms
Applied mathematics
Atoms & subatomic particles
Laboratories
Parity
ISSN:1064-8275, 1095-7197
Online Access:Get full text
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Summary:This paper describes a preconditioned conjugate gradient scheme for the Pn-h finite element discretization of a first-order system least-squares (FOSLS) formulation of the Boltzmann transport equation. The preconditioner is based on the norm equivalence between the FOSLS functional and a V norm. Its realization is an inexact inversion of the system of partial differential equations corresponding to this V norm. This preconditioner is essentially a moment-parity multigrid solver. It involves a 2 × 2 block diagonal system with each block describing only the interior like-parity coupling (even-even or odd-odd) of the spherical harmonic coefficients or moments. The interior cross-parity coupling is handled in the outer conjugate gradient iteration. Since the like-parity coupling consists of only second- and zeroth-order differential terms, whereas the full Pn system consists also of cross-parity, first-order coupling terms, the construction of a robust multigrid algorithm for each diagonal block is easier than the construction for the full Pn system. Numerical results indicate that this preconditioned conjugate gradient algorithm is more robust than a stand-alone multigrid solver for the full Pn system.
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ISSN:1064-8275
1095-7197
DOI:10.1137/S1064827502407172