This talk presents a posteriori error estimates as well as superconvergent postprocessing procedures for FE methods, both motivated by the smoothing passes in iterative linear solvers. We apply simple smoothers (one step of Jacobi/Gauss–Seidel sweep), on an auxiliary finer mesh to process the FE residual for a posteriori error control. For superconvergence, we use several smoothing passes for a higher-order FE linear system with the initial guess being a lower-order FE solution. The implementation is non-intrusive and is similar to a two-level iterative solver. Numerical examples include Poisson, biharmonic, Helmholtz and Maxwell equations. The material is based on joint works with Han Shui and Ludmil Zikatanov.