An Optimal Solver for Linear Systems Arising from Stochastic FEM Approximation of Diffusion Equations with Random Coefficients

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Abstract

This paper discusses the design and implementation of efficient solution algorithms for symmetric linear systems associated with stochastic Galerkin approximation of elliptic PDE problems with correlated random data. The novel feature of our preconditioned MINRES solver is the incorporation of error control in the natural “energy” norm in combination with a reliable and efficient a posteriori estimator for the PDE approximation error. This leads to a robust and optimally efficient stopping criterion: the iteration is terminated as soon as the algebraic error is insignificant compared to the approximation error. The MATLAB codes used in the numerical studies are available online.

Bibliographical metadata

Original languageEnglish
Pages (from-to)298-311
Number of pages14
JournalSIAM: Journal on Uncertainty Quantification
Volume4
Issue number1
Early online date13 Mar 2016
DOIs
StatePublished - 31 Mar 2016