  We investigate numerical solutions of high order curl problems with various
formulations and finite elements. We show that several classical conforming
finite elements lead to spurious solutions, while mixed formulations with
finite elements in complexes solve the problems correctly.
  To explain the numerical results, we clarify the cohomological structures in
high order curl problems by relating the partial differential equations to the
Hodge-Laplacian boundary problems of the gradcurl-complexes.
