Energy norm a posteriori error estimation for parametric operator equations

Bespalov, Alex and Powell, Catherine E. and Silvester, David (2014) Energy norm a posteriori error estimation for parametric operator equations. SIAM Journal on Scientific Computing, 36 (2). A339-A363. ISSN 1095-7197

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Official URL: http://dx.doi.org/10.1137/130916849

Abstract

Stochastic Galerkin approximation is an increasingly popular approach for the solution of elliptic PDE problems with correlated random data. A typical strategy is to combine conventional ($h$-) finite element approximation on the spatial domain with spectral ($p$-) approximation on a finite-dimensional manifold in the (stochastic) parameter domain. The issues involved in a posteriori error analysis of computed solutions are outlined in this paper. A novel energy error estimator that uses a parameter-free part of the underlying differential operator is introduced which effectively exploits the tensor product structure of the approximation space. We prove that our error estimator is reliable and efficient. We also discuss different strategies for enriching the approximation space and prove two-sided estimates of the error reduction for the corresponding enhanced approximations. These give computable estimates of the error reduction that depend only on the problem data and the original approximation.

Item Type: Article
Uncontrolled Keywords: stochastic Galerkin methods, stochastic finite elements, random data, Karhunen-Loeve expansion, parametric operator equations, error estimation, a posteriori error analysis
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 35 Partial differential equations
MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis
Depositing User: Alex Bespalov
Date Deposited: 14 Apr 2013
Last Modified: 20 Oct 2017 14:13
URI: http://eprints.maths.manchester.ac.uk/id/eprint/1967

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