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The Reduced Basis Method (RBM) is a rigorous model reduction approach forsolving parametrized partial differential equations. It identifies alow-dimensional subspace for approximation of the parametric solution manifoldthat is embedded in high-dimensional space. A reduced order model issubsequently constructed in this subspace. RBM relies on residual-based errorindicators or {\em a posteriori} error bounds to guide construction of thereduced solution subspace, to serve as a stopping criteria, and to certify theresulting surrogate solutions. Unfortunately, it is well-known that thestandard algorithm for residual norm computation suffers from prematurestagnation at the level of the square root of machine precision.  In this paper, we develop two alternatives to the standard offline phase ofreduced basis algorithms. First, we design a robust strategy for computation ofresidual error indicators that allows RBM algorithms to enrich the solutionsubspace with accuracy beyond root machine precision. Secondly, we propose anew error indicator based on the Lebesgue function in interpolation theory.This error indicator does not require computation of residual norms, andinstead only requires the ability to compute the RBM solution. Thisresidual-free indicator is rigorous in that it bounds the error committed bythe RBM approximation, but up to an uncomputable multiplicative constant.Because of this, the residual-free indicator is effective in choosing snapshotsduring the offline RBM phase, but cannot currently be used to certify errorthat the approximation commits. However, it circumvents the need for \textit{aposteriori} analysis of numerical methods, and therefore can be effective onproblems where such a rigorous estimate is hard to derive.

A robust error estimator and a residual-free error indicator for reduced basis methods