In the question Can the completion of a domain be a non-domain? it is shown that the completion of an integral domain could be not anymore a domain. I was wondering if there are conditions that ensure the completion stays an integral domain. For example if we consider $R$ a noetherian integral domain and we have that the ideal $I$ is generated by a regular sequence is then $R^{\wedge}_I$ a domain? I could not find much in the literature about topic.
I do not have a deep background in algebra, so I could not think of any approach to tackle such problem. Any contribution is welcomed.