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### Abstract

Reduced basis approximations for geometrically parametrized advection- diffusion equations are investigated. The parametric domains are assumed to be images of a reference domain through a piecewise polynomial map; this may lead to nonaffinely parametrized diffusion tensors that are treated with an empirical interpolation method. An a posteriori error bound including a correction term due to this approximation is given. Results concerning the applied methodology and the rigor of the corrected error estimator are shown for a shape optimization problem in a thermal flow.