A GENERAL OUTPUT BOUND RESULT: APPLICATION TO DISCRETIZATION AND ITERATION ERROR ESTIMATION AND CONTROL

Abstract

We present a general adjoint procedure that, under certain hypotheses, provides inexpensive, rigorous, accurate, and constant-free lower and upper asymptotic bounds for the error in "outputs" which are linear functionals of solutions to linear (e.g. partial-differential or algebraic) equations. We describe two particular instantiations for which the necessary hypotheses can 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 partial differential equations. The second case — new to this paper — assesses the error due to solution, in particular, incomplete iteration: a primal-dual preconditioned conjugate-gradient Lanczos method for symmetric positive-definite linear systems, in which the error bounds for the output serve as stopping criterion; numerical results are presented for additive-Schwarz domain-decomposition-preconditioned solution of a spectral element discretization of the Poisson equation in three space dimensions. In both instantiations, the computational savings are significant: since the error in the output of interest can be precisely quantified, very fine meshes, and extremely small residuals, are no longer required to ensure adequate accuracy; numerical uncertainty, though certainly not eliminated, is greatly reduced.