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

The backward error analysis is of the great importance in the analysis of 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 backward perturbations optimal with respect to the energy norm 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.