Quasi-monotonicity and Robust Localization with Continuous Piecewise Polynomials

We consider the energy norm arising from elliptic problems with discontinuous piecewise constant diffusion. We prove that under the quasi-monotonicity property on the diffusion coefficient, the best approximation error with continuous piecewise polynomials is equivalent to the $\ell_2$-sum of best errors on elements, in the spirit of A. Veeser for the $H^1$-seminorm. If the quasi-monotonicity is violated, counterexamples show that a robust localization does not hold in general, neither on elements, nor on pairs of adjacent elements, nor on stars of elements sharing a common vertex.