A general output bound result: Application to discretization and iterationerror estimation and control

We present a general adjoint procedure that, under certain hypotheses, prov ides inexpensive, rigorous, accurate, and constant-free lower and upper asy mptotic bounds for the error in "outputs" which are linear functionals of s olutions to linear (e.g. partial-differential or algebraic) equations. We d escribe two particular instantiations for which the necessary hypotheses ca n be readily verified. The first case - a re-interpretation of earlier work - assesses the error due to discretization: an implicit Neumann-subproblem finite element a posteriori technique applicable to general elliptic parti al differential equations. The second case - new to this paper - assesses t he error due to solution, in particular, incomplete iteration: a primal-dua l preconditioned conjugate-gradient Lanczos method for symmetric positive-d efinite linear systems, in which the error bounds for the output serve as s topping criterion; numerical results are presented for additive-Schwarz dom ain-decomposition-preconditioned solution of a spectral element discretizat ion of the Poisson equation in three space dimensions. In both instantiatio ns, the computational savings are significant: since the error in the outpu t of interest can be precisely quantified, very fine meshes, and extremely small residuals, are no longer required to ensure adequate accuracy; numeri cal uncertainty, though certainly not eliminated, is greatly reduced.