In this paper, we first establish an $L^2$-type Dolbeault
isomorphism for logarithmic differential forms by H\"{o}rmander's
$L^2$-estimates. By using this isomorphism and the construction of
smooth Hermitian metrics, we obtain a number of new
%Akizuki-Kodaira-Nakano type
vanishing theorems for sheaves of logarithmic differential forms on
compact K\"ahler manifolds with simple normal crossing divisors,
which generalize several classical vanishing theorems, including
Norimatsu's vanishing theorem, Gibrau's vanishing theorem, Le
Potier's vanishing theorem and a version of the Kawamata-Viehweg
vanishing theorem.