Semistable Reduction of a Normal Crossing $\mathbb{Q}$-Divisor

In a previous work we have introduced the notion of embedded $\mathbf{Q}$-resolution, which essentially consists in allowing the final ambient space to contain abelian quotient singularities, and A'Campo's formula was calculated in this setting. Here we study the semistable reduction associated with an embedded $\mathbf{Q}$-resolution so as to compute the mixed Hodge structure on the cohomology of the Milnor fiber in the isolated case using a generalization of Steenbrink's spectral sequence. Examples of Yomdin-L\^{e} surface singularities are presented as an application.