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A blackbox reduced-basis output bound method for noncoercive linear problems

Patera, Adolf, Rovas Dimitrios, Maday, Yvon, 1957-

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Summary

We present a reduced-basis output bound method for noncoercive linear problems and linear output functionals. The method is based upon (i) an enriched reduced basis comprising not only primal and dual solutions but also infimizers of the inf-sup condition associated with the bilinear form, (ii) {\sl either} a Galerkin {\sl or} a (more stable) two-space minimum-residual discretization, and (iii) an output error estimator formed from the dual norms of the primal and dual residuals and an approximation to the inf-sup parameter. The necessary calculations are effected by a blackbox superposition approach which exploits special (affine) parametric structure in the underlying operator: the operation count for the ``on-line" stage of the computational procedure depends only on the dimension of the reduced-basis space and the complexity of the parametric representation. For the minimum-residual formulation we can prove the optimality of the output approximation as well as the optimality and asymptotic bounding property of the output error estimator; thanks to the latter, only a minimal number of modes may be {\sl (safely)} retained. Numerical results are presented for a particular application: the Helmholtz equation.

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