Robust Flux Reconstruction and a Posteriori Error Analysis for an Elliptic Problem with Discontinuous Coefficients

In this paper, we locally construct a conservative flux for finite element solutions of elliptic interface problems with discontinuous coefficients. Since the Discontinuous Galerkin method has built-in conservative flux, we consider in this paper the conforming finite element method and a special type of nonconforming method with arbitrary orders. We also perform our analysis based on Nitsche's method, which imposes the Dirichlet boundary condition weakly. The construction method is derived based on a mixed problem with one solution coinciding with the finite element solution and with the other solution being naturally used to obtain a conservative flux. We then apply the recovered flux to the a posteriori error estimation and prove the robust reliability and efficiency for conforming elements, under the assumption that the diffusion coefficient is quasi-monotone. Numerical experiments are provided to verify the theoretical results.