We study an elliptic interface problem with discontinuous diffusion coefficients on unfitted meshes using the CutFEM method. Our main contribution is the reconstruction of an element-wise conservative flux from the CutFEM solution and its use in a posteriori error estimation. We introduce a hybrid mixed formulation with locally computable Lagrange multipliers and reconstruct an equilibrated flux in the immersed Raviart-Thomas space. Based on this, we propose a new a posteriori error estimator that includes both volume and interface terms. We state its robust reliability and local efficiency, which are proved in Part II of this work. The approach is validated through numerical experiments.
Capatina et al. (Tue,) studied this question.