Is the divorce that occurred between the spouses valid according to Sharia, considering that the husband did not utter the divorce formula, and that the divorce took place in the absence of both spouses, as the judge divorced her after the husband's approval?

If a judge rules for divorce due to harm inflicted upon the wife, the divorce is valid even if the husband does not pronounce it. It is not a condition for the divorce to occur that the wife be present; rather, it suffices for the husband to pronounce it verbally, or write it explicitly, or deputize someone else to divorce her.