  We propose a new algorithm for Adaptive Finite Element Methods (AFEMs) based
on smoothing iterations (S-AFEM), for linear, second-order, elliptic partial
differential equations (PDEs). The algorithm is inspired by the ascending phase
of the V-cycle multigrid method: we replace accurate algebraic solutions in
intermediate cycles of the classical AFEM with the application of a
prolongation step, followed by the application of a smoother. Even though these
intermediate solutions are far from the exact algebraic solutions, their
a-posteriori error estimation produces a refinement pattern that is
substantially equivalent to the one that would be generated by classical AFEM,
at a considerable fraction of the computational cost.
  We provide a qualitative analysis of how the error propagates throughout the
algorithm, and we present a series of numerical experiments that highlight the
efficiency and the computational speedup of S-AFEM.
