  Online enrichment is the extension of a reduced solution space based on the
solution of the reduced model. Procedures for online enrichment were published
for many localized model order reduction techniques. We show that residual
based online enrichment on overlapping domains converges exponentially.
Furthermore, we present an optimal enrichment strategy which couples the global
reduced space with a local fine space. Numerical experiments on the two
dimensional stationary heat equation with high contrast and channels confirm
and illustrate the results.
