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Multigrid transfers for nonsymmetric systems based on Schur complements and Galerkin projections
Author(s) -
Wiesner T. A.,
Tuminaro R. S.,
Wall W. A.,
Gee M. W.
Publication year - 2014
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.1889
Subject(s) - multigrid method , schur complement , mathematics , transfer operator , discretization , linear system , rate of convergence , galerkin method , grid , simple (philosophy) , key (lock) , finite element method , pure mathematics , mathematical analysis , computer science , geometry , partial differential equation , eigenvalues and eigenvectors , philosophy , physics , computer security , epistemology , quantum mechanics , thermodynamics
SUMMARY A framework is proposed for constructing algebraic multigrid transfer operators suitable for nonsymmetric positive definite linear systems. This framework follows a Schur complement perspective as this is suitable for both symmetric and nonsymmetric systems. In particular, a connection between algebraic multigrid and approximate block factorizations is explored. This connection demonstrates that the convergence rate of a two‐level model multigrid iteration is completely governed by how well the coarse discretization approximates a Schur complement operator. The new grid transfer algorithm is then based on computing a Schur complement but restricting the solution space of the corresponding grid transfers in a Galerkin‐style so that a far less expensive approximation is obtained. The final algorithm corresponds to a Richardson‐type iteration that is used to improve a simple initial prolongator or a simple initial restrictor. Numerical results are presented illustrating the performance of the resulting algebraic multigrid method on highly nonsymmetric systems. Copyright © 2013 John Wiley & Sons, Ltd.