Volume 40, pp. 338-355, 2013.

Energy backward error: interpretation in numerical solution of elliptic partial differential equations and behaviour in the conjugate gradient method

Serge Gratton, Pavel Jiránek, and Xavier Vasseur

Abstract

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.

Full Text (PDF) [349 KB]

Key words

symmetric positive definite systems, elliptic problems, finite element method, conjugate gradient method, backward error

AMS subject classifications

65F10, 65F50

Links to the cited ETNA articles

[2]Vol. 33 (2008-2009), pp. 31-44 M. Arioli and I. S. Duff: Using FGMRES to obtain backward stability in mixed precision

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