\begin{abstract} Let $R$ be a commutative Noetherian ring, $\fa\subseteq\fb$ two ideals of $R$, and $M$ a finitely generated $R$-module. We prove that $H_{\fb}^j(H_{\fa}^i(M))$ is $\fb$-cofinite for all $i$ and $j$ in the following cases: (1) $\Dim R/{\fb}=0$ and $\Dim R/{\fa}=1$ (2) $\Dim R/{\fb}=1$ and $\Dim R/{\fa}=1$. In case (1), we also prove that $H_{\fb}^j(H_{\fa}^i(M))$ is Artinian for all $i$ and $j$. Additionally, we show that if $\Dim R/{\fb}=1$ and $\Dim R/{\fa}=2$ and $n$ is a non-negative integer such that $H_{\fa}^i(M)$ is finitely generated for all $i
