  We prove a convergence result for a family of Yang-Mills connections over an
elliptic $K3$ surface $M$ as the fibers collapse. In particular, assume $M$ is
projective, admits a section, and has singular fibers of Kodaira type $I_1$ and
type $II$. Let $\Xi_{t_k}$ be a sequence of $SU(n)$ connections on a principal
$SU(n)$ bundle over $M$, that are anti-self-dual with respect to a sequence of
Ricci flat metrics collapsing the fibers of $M$. Given certain non-degeneracy
assumptions on the spectral covers induced by $\bar\partial_{\Xi_{t_k}}$, we
show that away from a finite number of fibers, the curvature $F_{\Xi_{t_k}}$ is
locally bounded in $C^0$, the connections converge along a subsequence (and
modulo unitary gauge change) in $L^p_1$ to a limiting $L^p_1$ connection
$\Xi_0$, and the restriction of $\Xi_0$ to any fiber is $C^{1,\alpha}$ gauge
equivalent to a flat connection with holomorphic structure determined by the
sequence of spectral covers. Additionally, we relate the connections
$\Xi_{t_k}$ to a converging family of special Lagrangian multi-sections in the
mirror HyperK\"ahler structure, addressing a conjecture of Fukaya in this
setting.
