  In this paper, we provide an $L^2$ fine resolution of the prolongation of a
nilpotent harmonic bundle in the sense of Simpson-Mochizuki (an analytic
analogue of the Kashiwara-Malgrange filtrations). This is the logarithmic
analogue of Cattani-Kaplan-Schmid's and Kashiwara-Kawai's results on the $L^2$
interpretation of the intersection complex. As an application, we give an
$L^2$-theoretic proof to the Nadel-Kawamata-Viehweg vanishing theorem with
coefficients in a nilpotent Higgs bundle.
