You will probably not incur problems -- after all, there are legions of people who have the same last name; you need only look in the telephone book under Smith to get hard evidence of this.

As long as your new mate's ex mate does not falsely hold herself out as still married to your man, any minor problems that may develop -- for example, your creditors' becoming confused -- should be easy to straighten out with a clear written explanation of who is and is not married to whom.