Optimal a priori estimates for interface problems

We consider a priori estimates in weighted norms for interface problems with piecewise constant diffusion constants which do not depend on the ratio between the constants. Our result generalizes an estimate of Lemrabet to arbitrary dimensions and includes curved boundaries. Furthermore, we discuss criteria for the existence of a uniform Poincaré estimate in weighted norms. In the affirmative case we obtain a robust finite element error bound in weighted norms. Finally, we present numerical experiments including a case with no uniform Poincaré constant.