  We observe a dramatic lack of robustness of the DPG method when solving
problems on large domains and where stability is based on a Poincar\'e-type
inequality. We show how robustness can be re-established by using appropriately
scaled test norms. As model cases we study the Poisson problem and the
Kirchhoff--Love plate bending model, and also include fully discrete variants
where optimal test functions are approximated. Numerical experiments for both
model problems, including an-isotropic domains and mixed boundary conditions,
confirm our findings.
