Robust a posteriori error control for transmission problems with sign-changing coefficients using localization of dual norms

We present a posteriori error analysis of diffusion problems where the diffusion tensor is not necessarily symmetric and positive definite and can in particular change its sign. We first identify the correct intrinsic error norm for such problems, including both conforming and nonconforming approximations. This involves global dual (residual) norms. Importantly, we show their equivalence with the Hilbertian sums of their localizations. We then design estimators which deliver simultaneously guaranteed error upper bound, local and global error lower bounds, and robustness with respect to the diffusion tensor as well as with respect to the approximation polynomial degree. The estimators are given in a unified setting covering at once conforming, nonconforming, mixed, and discontinuous Galerkin finite element discretizations. Numerical results illustrate the theoretical developments.