Locally computable error estimators for conforming approximations of interface problems cannot be robust

Abstract

We prove an impossibility result for extension-local a posteriori estimators for the conforming method for an elliptic interface problem at a checkerboard cross-point. We construct two diffusion problem instances on the same interface-fitted mesh that differ in the remote continuation of their coefficient-$M$ branches, yet every extension-local estimator satisfying efficiency must assign them the same value. They share a piecewise-constant load for which the finite element solution and the data oscillation both vanish. The ratio of their exact energy errors grows at least as the fourth root of the coefficient contrast $M$. Locality therefore forces the same estimator value for problems whose errors become increasingly different, yielding the central lower bound $C_{\rm eff}(M)C_{\rm rel}(M)\gtrsim M^{1/4}$, where $C_{\rm rel}(M)$ and $C_{\rm eff}(M)$ are the reliability and efficiency constants, respectively. Consequently, any locally computable error estimator, whether of residual, equilibrated, or recovery type, cannot be simultaneously reliable and efficient with contrast-independent constants, regardless of its algebraic form.