  Numerical computation of harmonic forms (typically called harmonic fields in
three space dimensions) arises in various areas, including computer graphics
and computational electromagnetics. The finite element exterior calculus
framework also relies extensively on accurate computation of harmonic forms. In
this work we study the convergence properties of adaptive finite element
methods (AFEM) for computing harmonic forms. We show that a properly defined
AFEM is contractive and achieves optimal convergence rate beginning from any
initial conforming mesh. This result is contrasted with related AFEM
convergence results for elliptic eigenvalue problems, where the initial mesh
must be sufficiently fine in order for AFEM to achieve any provable convergence
rate.
