Topics in matrix analysis
Convergence of iterations for linear equations
Convergence of iterations for linear equations
Alternating Direction Methods for Parabolic Systems in m Space Variables
Journal of the ACM (JACM)
Finite element solution of boundary value problems: theory and computation
Finite element solution of boundary value problems: theory and computation
Computational Differential Equations
Computational Differential Equations
Iterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems
An Algebraic Multigrid Method for Linear Elasticity
SIAM Journal on Scientific Computing
Convergence Analysis of a Multigrid Method for a Convection-Dominated Model Problem
SIAM Journal on Numerical Analysis
Stopping criteria for iterations in finite element methods
Numerische Mathematik
An overview of the Trilinos project
ACM Transactions on Mathematical Software (TOMS) - Special issue on the Advanced CompuTational Software (ACTS) Collection
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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 $\left(H^1\right)^2$.