  This paper uses convex integration to develop a new, general method for
proving relative $h$-principles for closed, stable, exterior forms on
manifolds. This method is applied to prove the relative $h$-principle for 4
classes of closed stable forms which were previously not known to satisfy the
$h$-principle, $\textit{viz.}$ stable $(2k-2)$-forms in $2k$ dimensions, stable
$(2k-1)$-forms in $2k+1$ dimensions, $\widetilde{\mathrm{G}}_2$ 3-forms and
$\widetilde{\mathrm{G}}_2$ 4-forms. The method is also used to produce new,
unified proofs of all three previously established $h$-principles for closed,
stable forms, $\textit{viz.}$ the $h$-principles for closed stable 2-forms in
$2k+1$ dimensions, closed $\mathrm{G}_2$ 4-forms and closed
$\mathrm{SL}(3;\mathbb{C})$ 3-forms. In addition, it is shown that if a class
of closed stable forms satisfies the relative $h$-principle, then the
corresponding Hitchin functional (whenever defined) is necessarily unbounded
above.
  Due to the general nature of the $h$-principles considered in this paper, the
application of convex integration requires an analogue of Hodge decomposition
on arbitrary $n$-manifolds (possibly non-compact, or with boundary) which
cannot, to the author's knowledge, be found elsewhere in the literature. Such a
decomposition is proven in Appendix A.
