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Hodge dual
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Given a primal k-cochain $\phi$, the discrete Hodge star of $\phi$ (denoted by $\star \phi$ or $\phi ^*$) is defined as follows. For a given $k$-chain $a$, the dual to $\phi$ $(n - k)$-cochain is defined by its value on the dual cell $\star a$ by $$\frac{1}{|\star a|}<\star \phi, \star a> = \frac{1}{|a|}<\phi, a>$$ or $$\frac{1}{|a^*|} \phi ^*(a^*) = \frac{1}{|a|}\phi(a).$$ Here $|b|$ is the measure, i.e., the $i$-dimensional volume, of an $i$-cell/chain $b$ (the measure of a $0$-cell is assumed to be $1$).
Therefore, the Hodge star is a diagonal matrix whose entries are the ratios of primal and dual volumes.
In particular, for dimension $1$, this coefficient is $$\frac{|point|}{|edge|}=1/\Delta x.$$ For dimension 2 in a cubical complex, it's $$\frac{|point|}{|rectangle|} = \frac{1}{\Delta x \Delta y}.$$
This is the general $2$-demensional case:
