It depends exactly when and where we are talking about, as well as the social class of the woman and her husband. One thing people forget is that even within what we now think of as monolithic social and legal regimes of France and Germany, there was, right up until the early-to-mid 19th-century a very large regional diversity of social custom and local laws. The situation in England was difference from the situation in Normandy, which was different from the situation in Picardy or in Ile-de-France or in Lorraine or Burgundy (the latter were not "French" in any meaningful sense until well after the High Middle Ages). The legal and social situation were even more different in Southern France (Langduoc, Aquitane, etc) or in the Germanies (note the plural).

With those disclaimers out of the way, let's try to examine your question. There are really three different questions here:

1. Could a woman leave an abusive husband?
2. Could she remarry without divorcing the first husband?
3. Are there known cases of this sort of thing?

All these depend heavily on region, as I said, but also on social class. A peasant woman without any other close living relatives was in a very different situation and a noblewoman whose extended family might support her financial, socially, and emotionally, in claims against her spouse. Now, for some actual answers:

1. Yes a woman could pick up and leave her husband of her own accord, but she might find it difficult to support herself and any children she took with her, unless she had sympathetic relatives who would help her out.
2. Remarrying without an annulment was basically impossible. Prior to the Protestant Reformation divorce didn ot exist in Western Europe among Christian population (who constituted the vast, +95% majority of the non-noble population and 100% of the nobility -- Spain's sort of a weird exception to this, but that's a topic unto itself). Obviously, Herrin Catherina Von Luxemburg would be able to find a sympathetic priest/bishop/$other_clergyman to give her an annulment, more easily than Annelise the Washerwoman. On the other hand, the peasantry didn't always care very much about official church approval of their social lives -- as long as there were no living children, and local community sort of tacitly approved of the new arrangement, local lower-class women could have a suprising amount of social autonomy, albeit largely by virtue of "flying under the radar" rather than having explicit rights like modern women do.
3. Yes, there documented cases of exactly the sort of unhappy domestic situation you describe. I don't have my books in front of me right now, but I'll post links/sources tomorrow when I'm able to do so. In all of these cases, the woman's ability to leave an abusive spouse depended on two things: 1) The support of the woman's relatives; it's very hard to leave your husband when the alternative is starvation -- it's easier when your sister-in-law is there to provide to emotional support and your brother is willing to support your claims because he wants control of your husband's lands, which were part of your dowry anyway. 2) The husband being a general social fuckup in other ways as well. The local community will look askance at a woman who left her husband becuase he beat her occasionally, but was otherwise an upstanding member of the community, but might be sympathetic to a a woman whose husband didn't feed her or her kids on a regular basis, wasted all the money on booze and gambling, AND also beat her regularly.

What part of Western Europe? Different laws regarding domestic abuse and marriage or divorce, or even moving town, or how independent women could be financially.