Assuming that there is an $\omega_1$-dense $\sigma$-complete ideal, $I$, on $\omega_1$ (such that every positive set can be partitioned into $\omega_1$ positive sets), we can generalize Joel's construction to $\lambda=\omega_1$ and prove that this forcing collapses $2^{\aleph_1}$. 

Let $p = \langle p_\alpha \mid \alpha < \omega_1\rangle$ be a condition. Let $A$ be the dense set in $I^+$ of size $\omega_1$. Let $r\in V$ be a real that codes an index of some $C\in A$, and $n<\omega$ such that $B=\{\alpha\in C\mid n\in\text{dom }p_\alpha\}\in I$. There is such $C, n$ since $I$ is $\sigma$-complete. 

Split $C\setminus B$ into $\omega_1$ sets from $A$, up to an error in $I$, $\{F_i \mid i < \omega_1\}$. We assume that the collection $\{F_i \mid i < \omega_1\}$ is the first such partition in some fixed well ordering. We can assume that for every $i < j < \omega_1$, $F_i \cap F_j = \emptyset$  (using the $\sigma$-completeness). Code an element of $y \in \mathcal P(\omega_1)$ by coloring the components: $q\leq p$, $q_\alpha (n) = 1$ iff $\alpha \in F_i$ where $i \in y$. Now, every $y \in \mathcal{P}(\omega_1)$ is coded by a real from $V$ which in turn coded by a natural number. 

The decoding is as follows: Let $G$ be the generic filter. Let $g_\alpha \colon \omega \to \omega$ be the generic real defined by $G$ for $\alpha < \omega_1$. For every natural $m$ we check if $n$ codes a real number in the means of Joel's answer. Assume that it does, and let $r_m$ be this real. By fixing a surjection from $2^\omega$ onto $A \times \omega$, $r$ corresponds to a couple $(C_m, n_m)$. Let $\{ F^m_i \mid i < \omega_1\}$ be the partition as above. Let:
 $$y_m = \{i < \omega_1 \mid \{\alpha \in F_i \mid g_\alpha (n) = 0\} = F_i \mod I\}$$.
By density arguments, we know that for every $y\in \mathcal{P}(\omega_1)^V$ there is $m$ such that $y = y_m$. 

It is not clear at all if we actually need the large cardinals here. Also, it is not clear if it's possible to generalize this method. A direct attempt to generalize this method would require the existence of dense ideals on both $\omega_1$ and $\omega_2$ which is an open problem (see comments below).