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A general output bound result: application to discretization and iteration error estimation and control. (English) Zbl 1012.65110

Summary: 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.

MSC:

65N15 Error bounds for boundary value problems involving PDEs
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
65F10 Iterative numerical methods for linear systems
65F35 Numerical computation of matrix norms, conditioning, scaling
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References:

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