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dc.contributor.authorMacLachlan, Scott
dc.contributor.authorMadden, Niall
dc.date.accessioned2018-09-20T16:15:20Z
dc.date.available2018-09-20T16:15:20Z
dc.date.issued2013-01-01
dc.identifier.citationMacLachlan, Scott; Madden, Niall (2013). Robust solution of singularly perturbed problems using multigrid methods. SIAM Journal on Scientific Computing 35 (5), A2225-A2254
dc.identifier.issn1064-8275,1095-7197
dc.identifier.urihttp://hdl.handle.net/10379/12554
dc.description.abstractWe consider the problem of solving linear systems of equations that arise in the numerical solution of singularly perturbed ordinary and partial differential equations of reaction-diffusion type. Standard discretization techniques are not suitable for such problems and, so, specially tailored methods are required, usually involving adapted or fitted meshes that resolve important features such as boundary and/or interior layers. In this study, we consider classical finite difference schemes on the layer adapted meshes of Shishkin and Bakhvalov. We show that standard direct solvers exhibit poor scaling behavior, with respect to the perturbation parameter, when solving the resulting linear systems. We propose and prove optimality of a new block-structured preconditioning approach that is robust for small values of the perturbation parameter, and compares favorably with standard robust multigrid preconditioners for these linear systems. We also derive stopping criteria which ensure that the potential accuracy of the layer-resolving meshes is achieved.
dc.publisherSociety for Industrial & Applied Mathematics (SIAM)
dc.relation.ispartofSIAM Journal on Scientific Computing
dc.rightsAttribution-NonCommercial-NoDerivs 3.0 Ireland
dc.rights.urihttps://creativecommons.org/licenses/by-nc-nd/3.0/ie/
dc.subjectboundary-fitted meshes
dc.subjectrobust multigrid
dc.subjectpreconditioning
dc.subjectstrongly discontinuous coefficients
dc.subjectreaction-diffusion problem
dc.subjectfinite-difference
dc.subjectshishkin meshes
dc.subjectnumerical-method
dc.subjectgrids
dc.subjectpreconditioner
dc.subjectconvergence
dc.subjectequations
dc.subjectsystem
dc.titleRobust solution of singularly perturbed problems using multigrid methods
dc.typeArticle
dc.identifier.doi10.1137/120889770
dc.local.publishedsourcehttp://neumann.math.tufts.edu/~scott/research/blpcg-TR.pdf
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