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From Functional Analysis to Iterative Methods

Published: 01 May 2010 Publication History
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  • Abstract

    We examine condition numbers, preconditioners, and iterative methods for finite element discretizations of coercive PDEs in the context of the fundamental solvability result, the Lax-Milgram lemma. Working in this Hilbert space context is justified because finite element operators are restrictions of infinite-dimensional Hilbert space operators to finite-dimensional subspaces. Moreover, useful insight is gained as to the relationship between Hilbert space and matrix condition numbers, and translating Hilbert space fixed point iterations into matrix computations provides new ways of motivating and explaining some classic iteration schemes. In this framework, the “simplest” preconditioner for an operator from a Hilbert space into its dual is the Riesz isomorphism. Simple analysis gives spectral bounds and iteration counts bounded independent of the finite element subspaces chosen. Moreover, the abstraction allows us to consider not only Riesz map preconditioning for convection-diffusion equations in $H^1$ but also operators on other Hilbert spaces, such as planar elasticity in $\left(H^1\right)^2$.

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        Published In

        cover image SIAM Review
        SIAM Review  Volume 52, Issue 2
        May 2010
        152 pages

        Publisher

        Society for Industrial and Applied Mathematics

        United States

        Publication History

        Published: 01 May 2010

        Author Tags

        1. functional analysis
        2. iterative methods
        3. preconditioner

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        • (2020)Product Algebras for Galerkin Discretisations of Boundary Integral Operators and their ApplicationsACM Transactions on Mathematical Software10.1145/336861846:1(1-22)Online publication date: 20-Mar-2020
        • (2019)Decomposition into subspaces preconditioning: abstract frameworkNumerical Algorithms10.1007/s11075-019-00671-483:1(57-98)Online publication date: 4-Feb-2019
        • (2015)Finite element approximation and preconditioners for a coupled thermal-acoustic modelComputers & Mathematics with Applications10.1016/j.camwa.2015.09.00470:10(2342-2354)Online publication date: 1-Nov-2015
        • (2015)Applying GMRES to the Helmholtz equation with shifted Laplacian preconditioningNumerische Mathematik10.1007/s00211-015-0700-2131:3(567-614)Online publication date: 1-Nov-2015

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