Geodesic
Energy backward error: interpretation in numerical solution of elliptic partial differential equations and behaviour in the conjugate gradient method
Electronic transactions on numerical analysis, Tome 40 (2013), pp. 338-355
Backward error analysis is of great importance in the analysis of the numerical stability of algorithms in finite precision arithmetic, and backward errors are also often employed in stopping criteria of iterative methods for solving systems of linear algebraic equations. The backward error measures how far we must perturb the data of the linear system so that the computed approximation solves it exactly. We assume that the linear systems are algebraic representations of partial differential equations discretised using the Galerkin finite element method. In this context, we try to find reasonable interpretations of the perturbations of the linear systems which are consistent with the problem they represent and consider the optimal backward perturbations with respect to the energy norm, which is naturally present in the underlying variational formulation. We also investigate its behaviour in the conjugate gradient method by constructing approximations in the underlying Krylov subspaces which actually minimise such a backward error.
Classification : 65F10, 65F50
Keywords: symmetric positive definite systems, elliptic problems, finite element method, conjugate gradient method, backward error 
@article{ETNA_2013__40__a8,
     author = {Gratton,  Serge and Jir\'anek,  Pavel and Vasseur,  Xavier},
     title = {Energy backward error: interpretation in numerical solution of elliptic partial differential equations and behaviour in the conjugate gradient method},
     journal = {Electronic transactions on numerical analysis},
     pages = {338--355},
     year = {2013},
     volume = {40},
     zbl = {1290.65029},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ETNA_2013__40__a8/}
}
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AU  - Jiránek,  Pavel
AU  - Vasseur,  Xavier
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JO  - Electronic transactions on numerical analysis
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%A Vasseur,  Xavier
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%J Electronic transactions on numerical analysis
%D 2013
%P 338-355
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%U http://geodesic.mathdoc.fr/item/ETNA_2013__40__a8/
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%F ETNA_2013__40__a8
Gratton,  Serge; Jiránek,  Pavel; Vasseur,  Xavier. Energy backward error: interpretation in numerical solution of elliptic partial differential equations and behaviour in the conjugate gradient method. Electronic transactions on numerical analysis, Tome 40 (2013), pp. 338-355. http://geodesic.mathdoc.fr/item/ETNA_2013__40__a8/