  This article studies a class of Dirac operators of the form $D_\varepsilon=
D+\varepsilon^{-1}\mathcal A$, where $\mathcal A$ is a zeroth order
perturbation vanishing on a subbundle. When $\mathcal A$ satisfies certain
additional assumptions, solutions of the Dirac equation have a concentration
property in the limit $\varepsilon\to 0$: components of the solution orthogonal
to $\ker(\mathcal A)$ decay exponentially away from the locus $\mathcal Z$
where the rank of $\ker(\mathcal A)$ jumps up. These results are extended to a
class of non-linear Dirac equations. This framework is then applied to study
the compactness properties of moduli spaces of solutions to generalized
Seiberg-Witten equations. In particular, it is shown that for sequences of
solutions which converge weakly to a $\mathbb Z_2$-harmonic spinor, certain
components of the solutions concentrate exponentially around the singular set
of the $\mathbb Z_2$-harmonic spinor. Using these results, the weak convergence
to $\mathbb Z_2$-harmonic spinors proved in existing convergence theorems is
improved to $C^\infty_{loc}$.
